{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "97cfff1b-1ccc-42d4-89e6-fee188714604",
   "metadata": {},
   "outputs": [],
   "source": [
    "####################################\n",
    "# This Sage worksheet is written by Bryan Curtis and Leslie Hogben hogben@aimath.org \n",
    "# using code written by numerous various people.\n",
    "# It includes computations for the paper\n",
    "# Zero forcing irredundant sets\n",
    "# by\n",
    "# Bryan Curtis, Leslie Hogben, and Riana Roux\n",
    "# \n",
    "# To run computations, you must first enter the function F- cells at the end of this file \n",
    "###################################"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6afe0d86-1374-44f5-8621-044dc27bcd1f",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Verify ZIR(P_4) (Observation 3.4)\n",
    "# up_low_ZIR(G) returns [ZIR(G), zir(G)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "82cf9689-71f2-4d3e-80ed-7d92f9337b5a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "Graphics object consisting of 8 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "[2, 1]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g=graphs.PathGraph(4)\n",
    "show(g)\n",
    "up_low_ZIR(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3c64b6a0-2414-412e-b031-bbd252e25131",
   "metadata": {},
   "outputs": [],
   "source": [
    "# ZIRsets(G) redurns all maximal ZIr-sets of G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "3e209b82-b766-4e46-b966-339fa0f7812d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{1, 2}, {3}, {0}]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZIRsets(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ad478914-2fc8-491e-b8ff-8a5784b3cd87",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Verify zir(H(3,5)), Z(H(3,5)), Zbar(H(3,5)), ZIR(H(3,5)) (comment before Proposition 3.6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7ef4cc48-5a6a-4543-bc9e-d45459f4e052",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 20 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g=Graph({0:[1,2,3],4:[1,2,3,5],6:[5,7],8:[7]})\n",
    "show(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "31eceb8b-6e92-4554-b147-7c4f6e172bfa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 2]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "up_low_ZIR(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "34124890-afea-46a3-a19a-a1648bd74a55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3fb41029-ac30-422e-b853-04fd24220098",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zbar(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "708f255b-cc8e-4a76-8642-14344a4ee810",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Example 7.2: Verify that zir(G)=3 for the graph G in Figure 7.2 (the pentasun)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "4a83e676-9637-4357-8f13-86178aa59493",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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IzS8w6d65+QWYvmwVIr89isDAQK7TIZtRaTAU6Q3Hv01M0uv1hjLvXb1xE1Peeh9dJ8/Cs4uWwd5OiYRta+Hu0qbMtXq9AUdOnkK//uVvO0BUG+Li4vD4449j9OjRcHBwQGxsLL7//nv0L+f70N/fH5GRkdhz7AS8pgVhX1w8yvu+B/7+ft4XFw+vaUHYeywBgwcPxqZNmxAREVHbXxJRnah0222ZTNYbQNLeFUsxYcjAan+SfXHxmLh4OTp37owVK1Zg/PjxfKREtSYxMRFLly7FkSNH4O3tjeXLl2PkyJGQyR6+4/CDJ7gN9faCl8YDTRo64fa9XKRoL+Bo0t8nuPn5+mJLWBg8PDwwd+5cbN68GZs3b8aLL75YB18lUbWY5zwGO6XymGurFgPSosOUphxe8qDc/AJ4TQtC09Zt0LRZM8TExMDT0xNLly7Fs88+W24/OFF1nD59GkuXLsXBgwfRrVs3LFu2DOPHjzcpEB5UfOZzYkICUtPSoNPpoFKp0N3TE/19fMqc+SyEwCuvvIJ169Zh3bp1CA4ONueXRmQu5gkGmUymlstlaVNH+KnMcYjJjz/+iGXLluHw4cN45JFH8MYbb2Dy5MlQKk3e6JWolF9//RVvvfUWPv/8c2g0GrzzzjuYNKn8RWjVZUo3nRACr7/+OlauXImVK1fi3//+t9k+P5GZmOegHiFEutEoAqszMRe4bDUiD8cgPDy8ZGJu4MCB+Pbbb5GYmAgPDw8EBATg0Ucfxc6dO6HX6026NxEAnD9/Hs8//zw8PT3x008/ITw8HL/88gv8/f3NPhI15RcimUyGDz74AG+++SZee+01LF++3Kw1ENUZIYRJf6KiooSjo6PwcG0r9qx4UxQePySMCd+W+VN4/JDYs+JN4eHaVjg6Ooro6GhRmZMnT4rx48cLAMLDw0Ns27ZN6HS6Sj+G6rfLly+LOXPmCKVSKVxcXMSGDRtEQUGB1GWVsmzZMgFALF26VBiNRqnLISpm0s97k4NBCCG0Wq3wHTJEABCurVuJ558cJlb/a47YtmS+WP2vOeL5J4cJ19atBADh5+srtFqtydWePn1aTJw4UchkMuHm5iY2bdpkcf+xk7QyMzPFv/71L2Fvby9atmwpVq1aJXJzc6Uuq0IhISECgFi4cCHDgSyF+YOhWFJSkpg3b57o4+0tVCqVACBUKpXo4+0t5s2bJ5KSkqpddVpamvD39xcymUy4urqK0NBQkZ+fX+37kfW7efOmeP3110WDBg1EkyZNxPLly8WdO3ekLsskH374oQAgXn75ZYYDWYLaC4YHGQwGs1f/66+/imnTpgm5XC5cXFzEmjVrLPq3QzK/W7duibfeeks0atRIODk5iTfeeENkZ2dLXVaVbdq0SQAQL730Uq38t0JUBXUXDLVJq9WKGTNmCKVSKVq3bi1CQkLE3bt3pSqH6sC9e/fE+++/L5o3by5UKpV49dVXRVZWltRl1Uh4eLiQyWRixowZQq/XS10O1V+2EQzFLly4IObMmSPs7OxEixYtxHvvvSdu374tdVlkRvn5+eKjjz4Sbdq0EXZ2diIoKEhcuXJF6rLMZteuXUIul4uAgABRVFQkdTlUP9lWMBS7dOmSmDt3rrC3txfNmjUT77zzjsjJyZG6LKqBwsJCsWXLFtGuXTshl8tFYGCg+OOPP6Quq1bs3r1bKBQKMWnSJFFYWCh1OVT/2GYwFLty5Yp4+eWXhYODg2jcuLF48803xc2bN6Uui6pAr9eLjz/+WHh4eAiZTCb8/f3Fb7/9JnVZtW7//v3Czs5OjB8/np13VNdsOxiKZWZmigULFogGDRqIhg0bikWLFln982hbZzAYxGeffSYeeeQRAUCMHz9enDlzRuqy6tTBgweFSqUSo0ePZtcd1aX6EQzFsrKyxKJFi0TDhg1FgwYNxIIFC0RmZqbUZdF9jEajOHDggOjZs6cAIEaOHCl+/vlnqcuSzHfffSccHR3F8OHD2XFHdaV+BUOxmzdvijfffFM0btxYODg4iJdffllcvXpV6rLqNaPRKI4cOSL69+8vAIgnnnhC/PDDD1KXZRFiY2OFk5OTGDJkCLvtqC7Uz2AolpOTI9555x3RtGlToVKpRFBQkLh06ZLUZdU78fHxYsj/rZbv27ev+P7777nQ6wHx8fGiUaNGYuDAgey0o9pWv4Oh2O3bt8V7770nWrRoIezs7MScOXNstuPFkpw8eVKMGjVKABBeXl7iq6++YiBUIjExUTRp0kT07dvXKhfxkdVgMNzv7t27IiQkRLRq1UoolUoxY8aMKu3lRKZJTU0VEyZMEABEly5dxO7du7na10RJSUmiefPmolevXuywo9rCYChPbm6uWLNmjXB2dhZyuVxMmzatXrRI1rZz586V7HHVsWNHsXPnTi7iqoYzZ86IVq1aie7du4vr169LXQ7ZHgZDZfLy8kRoaKhwdXUt6aFPS0uTuiyrc/HiRTFjxgyhUCiEq6ur2Lx5M7dNr6GzZ88KZ2dn8cgjj4hr165JXQ7ZFgaDKQoKCsSmTZuEm5ubkMlkYuLEiSIlJUXqsize1atXxdy5c4WdnZ1o3bq1+PDDD9mPb0a///67cHV1FRqNRmRkZEhdDtkOBkNV6HQ6sW3bNtGxY8eSRVc12T7cVmVlZYkFCxYIBwcH0axZM7FixQq2WdaS8+fPC3d3d9GxY0c2TJC5mPTz/qFnPt9/2JtZj46zUEVFRYiOjsa7774LrVaLMWPGYOnSpejXr5/UpUnq1q1bWL16NT788EPIZDK8+uqrmD9/Ppo0aSJ1aTbt8uXL8PPzQ1FREWJiYtCpUyepSyLrZtKZzxwxVKCoqEhERkaKrl27lqzSjY+Pl7qsOnf37l2xfPly0bRpU+Ho6CgWLlwobty4IXVZ9cqVK1dEly5dRNu2bdkoQTXFR0nmoNfrxe7du4Wnp+ffR5b6+Ym4uDipy6p1eXl5YvXq1aJly5bC3t5eBAcHcyJUQpmZmeLRRx8Vbdq0EampqVKXQ9aLwWBOBoNB7N27t2SfnyeeeEIcOXLE5hZt6XQ6sXHjRtG2bVuhUCjE7NmzuWLcQmRlZQkvLy/RsmVLcfr0aanLIetk0s97eW0+zLIlcrkczzzzDJKTk/HVV18hLy8Pw4YNw8CBA/Htt9/+nbJWTK/XIyIiAp07d8bcuXPh5+eH3377DWFhYXBzc5O6PALQqlUrxMTEwN3dHb6+vjh58qTUJZGNYjBUkUwmw9ixY/Hzzz/jm2++gRACTz75JPr164eDBw9aXUAYjUZ8+umn6NatG2bMmIHHHnsMaWlp2LVrF9RqtdTl0QOaN2+Oo0ePokuXLhg6dCgSEhKkLolsEIOhmmQyGUaNGoUTJ07g+++/h4ODA8aOHQtvb2/s378fRqNR6hIrJYTAF198AS8vL/j7+0Oj0SApKQmff/45Hn30UanLo0o0adIE3333Hby8vDBixAj88MMPUpdENobBUEMymQzDhg3DDz/8gLi4ODRr1gzPPPMMevbsic8//9ysAWGOewkhcPjwYfTt2xcTJkxAmzZtcOLECRw8eBC9e/c2Q5VUFxo1aoRvvvkGffv2xZNPPomjR49KXRLZEAaDGQ0ePBhHjx7F8ePH4eLigkmTJqF79+6Ijo6GwWCo8v2Sk5MRHByMPt7ecHBwgEKhgIODA/p4eyM4OBjJyclVut8PP/yAwYMHY9SoUbCzs0NMTAyOHDkCHx+fKtdG0nNycsLBgwfxxBNP4KmnnsLhw4elLolshamz1HU3aW47EhMTxZgxYwQA0blzZ5M3ltNqtcL3/84wcG3dSrwwephY8/KLYtuS+WLNyy+KF0YPE66tWwkAwnfIkIfuEpuYmCiGDx8uAIjevXuLr7/+2ua6qeqzgoICMXbsWGFvby+++uorqcshy8Z2VUtx8uRJ8fTTTwsAwsPDQ4SHh4vCwsJyr42KihKOjo7Cw7Wt2LtiqSg8fkgYE74t86fw+CGxd8VS4eHaVjg6Ooro6Ogy9zp9+rQYO3asACAeffRRsXfvXgaCjdLpdOLZZ58VSqVS7NmzR+pyyHIxGCzNqVOnxLPPPisACHd3d7F582ZRUFBQ8n5UVJSQyWRi2qih4m7MF+UGwoN/7sZ8IaaNGipkMpmIiooSQgjx66+/ikmTJgkAolOnTiIyMlLo9XqpvmyqI0VFReK5554TCoWi3F8UiASDwXKlpqaK5557TshkMtGuXTsRGhoqUlNThaOjo5g2aqjQ//h1qR/+6/89V3RwaSNU9naidxe1OLZpVan39T9+LaaNGiocHBzE+PHjhVwuF25ubmLbtm0VjkzINun1evH8888LuVwuduzYIXU5ZHlM+nnPTfQk9Ntvv+Hdd99FdHQ0VPb2cG7eDGciN8HJ0aHkmt1HjuH5d1Ziw2tzMbBHN2zZ/zXCD3yLs9FhcHNuXXJdbn4BuvnPQdbtO1i5ciVmz54NlUolxZdFEjMajXjxxRcRHh6OsLAwzJo1S+qSyHKYtIkeu5Ik1LVrV+zatQt79uxBfkEBVgXPKhUKALD2k32YMXYkZo17Eo90cMOH819C+9atsGnfwVLXOTk6YM0rc1BQUIABAwYwFOoxuVyOLVu2ICgoCLNnz8aGDRukLomsjFLqAgiIiYlBuzatMXZQ/1KvFxYVIel3LV6fNqnU68P79UZC6q9l7jNukA9cW7dCREQE1yTUc3K5HKGhoVCpVJg3bx50Oh1effVVqcsiK8FgsAAJJ05gqHcPKJWKUq/fvHUHBoMRbZo3K/V6m2bN8Gd2dpn7KJUKDPX2QiK3SSD8vfhy1apVUKlUWLBgAXQ6HRYvXix1WVTLjEYj5PKaPQzioyQLkHb2LLw0FR/AInvgqaCAgKyCR4VeGg+kpqWZszyyYjKZDO+++y7efvttLFmyBO+8847V7edFlTP3QliAIwbJGY1G6HQ6NHZqUOa9lk0bQ6GQ48+/ckq9npVzq8wooliThk7Q6XRm+a2BbINMJsNbb70FlUqFxYsXQ6fT4d1334Xswd84yKqkp6djzuzZiI2Lg2vrVhjWxwtTH5+Oxk4NcCc3Dyna89j/2W6sX78evkOGIGzrVpM3xmQwSEwul0OlUuFObl6Z9+zt7ODdRYPv/3cKE4YMLHn9yM+nMO7x/mWuB4Db93KhUqkYClTGokWLoFKp8Oqrr6KgoACrV69mOFip6OhozJo1Cy7Nm2HviqUYO6h/mUfRAKDXG3AgPhGvrQ9Hjx49kJ+f7y+E+ORh92cwWADPbt2Qoj1f7nvz/Z/B8++sRJ+uGvh0fwRhX3yDy9ez8NKEMeVen6K9gO6enrVZLlmx+fPnw97eHvPmzUNhYSHWrVvHXyKsTHR0NAICAhAw0g8bFwaX6WS8n1KpwIQhAzGinzeCQkKx69ujUTKZTCaEiK7sczAYLIDPgAHY/9lu6PWGMqk/edhg/HX7DpZtj0LmXznw9HDHodXL4O7Spsx99HoDjialYMKkyXVVOlmhuXPnwt7eHi+++CJ0Oh22bNnCcLASWq0Ws2bNQsBIP0QsXQC5XA693oC3w3ch+nAs/vwrBy4tm+OF0cPxZqB/yd+rk6MDIpYugAAQ/V3MdplM9rMQIr2iz8MFbhYgOTkZ3t7e2LtiaalHRlW1Ly4eExcvR1JSEttV6aF27tyJGTNmICAgANu3b4dCUfZRBFkWP19fXE4/h9MfbywZKby74xN8+Ol+7Fi6AN083HHyVy1mvLsGy+a8gJcnjy/18bn5BfCcMkd/9cZfJ4r0+sEVfR7+mmABevfuDd8hQ/Da+nDk5hdU6x65+QVYuH47fIcMYSiQSV544QVERUUhKioKAQEBKCoqkrokqkRSUhJi4+IQMndmqcdHiam/Ytzj/TFmYD90cHHGRL/HMaJvbyT9dq7MPZwcHbD65TlKvcHwhEwmq/AHBYPBQoRt3YrM7BwEhYRW+UAeo9GIoJBQZGbnIGzr1lqqkGzRc889h927d2PPnj147rnnUFhYKHVJVIEdO3aUuxB2oFc3xJw8jXOXrwD4e54xPuUsnvR5rNz7jBvkgzbNm+kBBFb0uTjHYCHUajXCw8MxdepUAHjopFKx3PwCBIWEIvJwDKKionhOM1XZs88+i3379mHixImYOHEiPv/8c26pYoEqWgj7+rRJuH0vF488NxsKuRwGoxHLX3wB/iN8y72PUqnAyH7eyk++ix1U0efiiMGC+Pv7IzIyEnuOnYDXtCDsi4uHXl/+yW96vQH74uLhNS0Ie46dQFRUFPz9/eu4YrIVY8eOxZdffonvv/8eTz/9NPLz86UuiR5Q0ULY3UeOIepwDKLeeR1JO9Zjx9IFWB29FzsPfV/hvbw0HjAYjd0qep8jBgszZcoU9O3bF3Nmz8bExcvh2roVhnp7wUvjgSYNnXD7Xi5StBdwNCkFV7NuwM/XF4fDwjhSoBobNWoUDh06hLFjx2LMmDE4cOAAnJycpC6LUPlC2IXrt+H1aZPw3PAhAIDu6o649GcW3v94N14YM7zc+zVp6ASjEHYymUwuhCjz7JrBYIHUajViYmORnJyMiIgIJCYkYHfMTuh0OshlMnTu3BkTJk1GYGAgJ5rJrPz8/PDtt99i9OjRePLJJ3Ho0CE0atRI6rLqHSEEMjIycPr0aZw6dQqnTp2CXC4vdyFsXoGuTLuxQi6HsZKO09v3ciGXyYoMFUxoMhgsWO/evUv94M/Pz0fDhg2x4N//5h77VGsef/xxfPfddxg1ahRGjBiBb775Bk2bNpW6LJtlMBhw7ty5kgA4deoUTp8+jb/++gsA0LJlS/Tq1QttWrcudyHs2EH98N6OT+HWphW6ebjj1O/nsfbT/Qh8akSFnzNFewEKufxsRe9zHYOV6dixIyZNmoQPPvhA6lLIxp08eRIjRoyAh4cHvvvuOzRv3lzqkqxeQUEBUlNTS4XAmTNnSuZ0OnTogJ49e6JXr14lf1xdXSGTyRAcHIz9n+3GH3t3lJqAvpubh6VhH+OLH04gK/sW2rZqgeeGD8Z/ZkyFvZ1dmRr0egPaPx2gv56ds1kIEVxenQwGKzNixAg0atQIe/fulboUqgdOnz6N4cOHw9XVFd9//z1atWoldUlWIycnp9SjoFOnTuG3336DwWCAQqFA165dSwVAz5490axZ+ZtjAuZfCAvAWwhR7tarDAYrExQUhPj4eJw5c0bqUqieOHv2LIYOHYoWLVrg6NGjcHZ2lrokiyKEwNWrV8s8Crp48SIAwNHRET169CgVAp6ennB0dKzy5/Lz9cUl7Tmk7NpoUjv7g0xd+cxgsDJr167FG2+8gXv37nF/G6ozv//+O/z8/NCwYUPExMTA1dVV6pIkYTAYoNVqy4TAzZs3AQDNmzcvFQC9evVC586dzbbdSHp6Onr06IGJgweU7JVkKqPRiOnLVovo72IKjUbhyb2SbMjBgwcxduxYZGRkoF27dlKXQ/VIeno6/Pz8YG9vj5iYGLi5uT30Y6z5XJCCggKkpaWVCoCUlBTk5f3dGeTm5lYmBNq1a1frW5l/8sknmDp1qkm7qxYrXgi769ujAsDUh229zWCwMr///ju6du2KmJgY+PqWv7KRqLZcvHgRfn5+MBqNiImJgYeHR6n3i1usE06cQNrZs9DpdFCpVPDs1g0+AwZYbIv1rVu3SuYDiv/566+/Qq/XQy6Xl5kP8PLyQosWLSSr9/7zGELmzcC4QT4VnsfwVXwCFq7fjszsHOTn508x5TwGBoOVKSwshKOjIzZv3ozZs2dLXQ7VQ1euXIGfnx/y8vIQExODzp07l3uamJemU6nTxI6c/HtRZlVPEzMnIQSuXbtWKgBOnTqFP/74AwDg4OBQMh9Q3B3UvXt3NGhQdmGZ1B78//xhC2G3/L0Q1qThDIPBCnl4eGDixIkICQmRuhSqpzIzMzF06FDk5ORg/vz5ePvtt+HSvBlWzptp0mlimdk5CA8Pr9VtXIxGY8l8wP0hcOPGDQBAs2bNSgVAr1690KVLFyiV1rW86/6FsKlpaSWjtO6enujv4/PgKI3BYKtGjhyJBg0aYP/+/VKXQvVYVlYW+vTpg4yMDASMGopN1dj4MTIyElOmTKlxLTqdDmlpaaUCICUlBbm5uQCA9u3bl1kf4ObmZpNHmz5kXsekL9i6opEAABqNBnFxcVKXQfXc7du3cfPGDUwd5Ycd/9chs2nfQWzedxAXM7MAAN083LB0xtRSW0AXnyYGALNmzULfvn2r9Fjp9u3bSElJKdUZ9Msvv5TMB3Tp0gU9e/bE+PHjS0YELVu2NO8Xb8HMMdnPEYMV+vDDD7F48WLk5uZabccHWb/yThM7cDwRCoUc6nZtAQA7vz6CVVF7kLxzPbp5dCj18bn5BfCaFgR3TWfExMaW+zkyMzNLBcCpU6dw4cIFAPj7cUn37qVGAd27d+fGf5XjoyRbdejQITz11FO4fPky2rdvL3U5VA8lJSWhT58+Jq3CbTFiIkLmzcLMcaPKvFe8Cvd///sfmjRpUmZ9wPXr1wEATZs2LfMoqGvXrlY3H2AB+CjJVmk0GgB/HwzOYCApVHSa2P0MBgM+jzmO3AIdfLo/Uu414wb5wLlFMwwYMKDkaFFXV1f06tULc+bMKQkBd3d3m5wPsFQMBivUoUMHKBQKaLVa+Pn5SV0O1UMVnSYGAKnpf2DAnPkoKCxEQ0dH7Ht/KR7t6F7ufZRKBUb09UZc2m/YFr4dPXv25H5MFoAPqK2Qvb093N3dkZ5e4Yp2olpV0WliANDFvR1O7dyIhK0f4qUJYzB92Wr88selCu/lpfHA9awbGD58OEPBQjAYrJRGo4FWq5W6DKqHKjtNDADs7eygbt8WfR7pjBVBM+Cl7oiPdn9R4f2aNHSCTqdDBWfGkAQYDFaKwUBSkcvlUKlU5Z4mVh4hgML/mz8oz+17uVCpVOywsyD8m7BSGo0G58+f529ZJAnPbt3KPU1syaYIHD+dhouZfyI1/Q+8sXkH4k6dwZSRFc+FpWgvoLunZ22WS1XEyWcrpVarodPpcOXKFZN2uSQyJ58BA7D/s93Q6w2lJqCvZ+fg+XdCkPlXDpo0bIAenTrim7XLMbxv+Rvn6fUGHE1KwYRJk+uqdDIBg8FK3d+yymCguhYYGIj169fjQHxiqXUM4W+8WqX7fBWfgKtZNxAYGGjuEqkG+CjJSt3fskpU13r37g3fIUPw2vpw5OYXVOseufkFWLh+O3yHDLHIrbjrMwaDlbKzs0PHjh0ZDCSZsK1bkZmdg6CQ0CrPdRmNRgSFhCIzOwdhW7fWUoVUXQwGK6ZWq7mWgSSjVqsRHh6OyMMxmL5stckjh9z8AgQuW43IwzEIDw+X5FwGqhyDwYqxZZWk5u/vj8jISHx29Ad085+DfXHx0OsN5V6r1xuwLy4eXtOCsOfYCURFRdXqeQxUfZx8tmIajQZbtmyBwWAw22HjRFU1evRo2KtUgEqFiYuXm3Sa2OG/TxOTunSqAIPBimk0GhQWFiIjIwMdOnSQuhyqpz766CPo9Xr89NPPuHbtWslpYrtjdpY6TWzCpMkWe+YzlcZgsGL3t6wyGEgKt27dwtq1a/HSSy/B2dkZzs7OpX7wP+Q0MbJQ/BuzYu7u7lAqlZyAJsl89NFH0Ol0WLhwYbnvMxSsE//WrJhSqWTLKknm/tGCi4uL1OWQGTEYrBw7k0gq69atq3S0QNaLwWDlGAwkBY4WbBuDwcqp1WpcuHABBkP5veNEtWHdunUoKCjgaMFGMRisnEajQVFRES5fvix1KVRPFI8WXnzxRY4WbBSDwcrd37JKVBeKRwuvv/661KVQLWEwWDk3NzfY2dkxGKhO3L59m6OFeoDBYOWKW1a5loHqAkcL9QODwQawM4nqwu3bt7FmzRqOFuoBBoMNYDBQXeBoof5gMNgAjUaDCxcuQK/XS10K2aji0cKcOXM4WqgHGAw2QK1WQ6/Xs2WVas26deuQn5/P0UI9wWCwAWxZpdp0fydS27ZtpS6H6gCDwQa4ubnB3t6ewUC1IjQ0FHl5eRwt1CMMBhugUCjg4eHBYCCzu78TiaOF+oPBYCPUajWDgcyOo4X6icFgIzQaDRe5kVnd34nE0UL9wmCwERqNBn/88QdbVslsOFqovxgMNkKj0UCv1+PixYtSl0I24M6dOyWjBVdXV6nLoTrGYLARbFklc+JooX5jMNiIdu3awd7envMMVGN37tzB6tWrOVqoxxgMNkKhUKBTp04cMVCNhYaGIjc3l6OFeozBYEO4mR7VFEcLBDAYbAqDgWqqeLSwaNEiqUshCTEYbIharcbFixdRVFQkdSlkhdiJRMUYDDZEo9HAYDCwZZWqZf369bh37x5HC8RgsCVsWaXq4twC3Y/BYEPatWsHlUrFYKAqKx4tsBOJAAaDTZHL5ejUqRPXMlCV3L17F6tXr8bs2bPRrl07qcshC8BgsDHsTKKq4twCPYjBYGMYDFQVd+/exapVqzhaoFIYDDZGo9Hg4sWLKCwslLoUsgIcLVB5GAw2Rq1Ww2g0smWVHoqjBaoIg8HGsGWVTMXRAlWEwWBjXF1d4eDgwGCgShWPFmbNmsXRApXBYLAxcrmc5z/TQ23YsIGjBaoQg8EGMRioMvePFtq3by91OWSBGAw2SKPRcJEbVWjDhg24e/cuRwtUIQaDDdJoNLh06RJbVqkMjhbIFAwGG6TRaGA0GnHhwgWpSyELs2HDBty5c4ejBaoUg8EGsWWVynPv3j2OFsgkDAYb5OLiAkdHR84zUCnFo4XFixdLXQpZOAaDDWLLKj3o3r17WLlyJUcLZBIGg43iZnp0P44WqCoYDDaKwUDFOFqgqmIw2Ci1Wo3Lly9Dp9NJXQpJjJ1IVFUMBhul0WgghGDLaj1X3Ik0c+ZMuLm5SV0OWQkGg41iyyoBwMaNG3H79m3OLVCVMBhslIuLCxo0aMBgqMeK5xY4WqCqYjDYKJlMBrVazbUM9RhHC1RdDAYbxs6k+oujBaoJBoMNYzDUXxwtUE0wGGyYRqNBRkYGCgoKpC6F6lBubi5WrlyJGTNmcLRA1cJgsGFqtZotq/UQRwtUUwwGG8aW1fonNzcXISEhmDFjBtzd3aUuh6wUg8GGOTs7o2HDhgyGeoSjBTIHBoMNK25ZZTDUDxwtkLkwGGwcg6H+4GiBzIXBYOM0Gg0XudUDxZ1IgYGBHC1QjTEYbFxxy2p+fj6MRqPU5VAt2bRpE3JycrBkyRKpSyEbwGCwYcnJyTh48CCUCgWaNm0KhUIBBwcH9PH2RnBwMJKTk6UukcyAcwtkbjIhhKnXmnwhSSs9PR1zZs9GbFwcXFu3xLA+PeGl6YTGTg1wJzcPKdrzOHIyBVezbsB3yBCEbd0KtVotddlUTatWrcLixYuRnp7OYKCHkZl0EYPBtkRHR2PWrFlwad4MK+fNxNhB/aFUKspcp9cbcCA+Ea+tD0dmdg7Cw8Ph7+8vQcVUE7m5uejYsSMmTJiALVu2SF0OWT6TgkFZ21VQ3YmOjkZAQAACRvph48JgODk6VHitUqnAhCEDMaKfN4JCQjF16lQIITBlypQ6rJhqqnhugZ1IZE4cMdgIrVYLLy8vTBw8ABFLF0Au/3v66IdTqVgVtQdJv2uReTMb+97/D8YPHlDqY41GIwKXrcaeYydw5swZPlayErm5ufDw8MDTTz+NsLAwqcsh62DSiIGTzzbixTlz0LZFM2xcGFwSCgCQW1CAHpqOCF0QVOHHyuVybFwYDJfmzTBn9uy6KJfMYPPmzcjOzmYnEpkdHyXZgKSkJMTGxWHviqVlHh896fMYnvR57KH3cHJ0QMi8GZi4eDmSk5PRu3fv2iqXzKC4EykwMBAdOnSQuhyyMRwx2IAdO3agXZvWGDuof43uM26QD1xbt0JERISZKqPawtEC1SYGgw1IOHECQ717lNt9VBVKpQJDvb2QmJBgpsqoNuTl5XG0QLWKwWAD0s6ehZemk1nu5aXxQGpamlnuRbWDowWqbZxjsHJGoxE6nQ6NnRqY5X5NGjpBp9Nh/fr16NChA9zd3eHm5oYmTZqY5f5UM3l5efjggw8wffp0jhao1jAYrJxcLodKpcKd3Dyz3O/2vVzI5XLMnz8fer2+5PXGjRuXhER5/3R2doZCUbNHWfRwHC1QXWAw2ADPbt2Qoj1f7nv38vKRfuVayb//ce1PnD53Hs0bN4Kbc+sy16doL6B3r15I/OknXL9+HZcuXcLly5dL/TM+Ph6XLl3C7du3Sz5OqVSiffv2cHNzKzc83Nzc0KCBeUY19YnRaCxpP75/tNCxY0eJKyNbxgVuNiA4OBj7P9uNP/buKDMBHZecAr+5r5f5mBdGD0PE0n+Xek2vN6Djs9MxYdJkhIaGPvTz3r59G5cvXy7582CIXLt2rdSOri1btiw3MIr/d6tWrSCTmbT+xmYlJycjIiICCSdOIO3sWeh0OqhUKnh26wYHR0ckJiZCq9UyGKi6uFdSfZGcnAxvb2/sXbEUE4YMrPZ99sXFY+Li5UhKSjLLOoaioiJcvXq13FFH8T/z8/NLrndwcKjwUZWbmxvatWsHe3v7GtdliUpvfNgKw/p4ldn48NvEJFzPzuHGh1QTDIb6ZPATT+CS9hzSordUukdSRXLzC+A1LQjums6IiY2thQrLEkLgr7/+qjA0Ll++jKysrJLrZTIZXFxcKg2Ppk2b1knt5sSND6kOMRjqi6tXr2L48OH4/fffMWWEL3bct1eSKSx5r6T8/HxkZGRUGB4ZGRkoKioqub5x48YVznO4u7vDxcXFoibJq7LxYbHc/AIEhYQi8nAMIiMjufEhVQWDoT5ISUnBmDFjIJfLMW/ePCxatKjaP2SioqKs7jdQo9GIP//8s1RgPBget27dKrleqVSiXbt2FU6Qu7m5wcnJqU5qL2/jwxU7P8X+Yz/it0tX4Kiyx4Duj+L9oBno4t6+zNdtqWFOFo3BYOu+/vprTJ48GV26dMGBAwfg4uJS6rFEyLwZGDfIp8LHEl/FJ2Dh+u02/1jizp07FU6QX758GVevXi01Sd6iRYtKW3PNNUnu5+uLy+nncPrjjSUh/uQrb2Dy8MF47JHO0BuMeHPzDqReuIiz0WFlgl6Kx39k9RgMtmzTpk2YN28ennrqKURHR5f6LffBicyh3l7w0nigSUMn3L6XixTtBRxN+vsENz9fX2wJC6vXv3EWFRXh2rVrlU6S5+X9/3UiDg4OaN++fYXh0b59+4dOkiclJaFPnz4PbRi4kXMLbUY/h7iNK/FEr+5l3jd3wwDZPB7UY4sMBgMWLlyINWvW4OWXX8bq1avLPDNXq9WIiY0taX1MTEjA7pidJa2P3T09MWHSZAQGBvKHCQA7Ozu4u7tXeCymEALZ2dnlhkZqaioOHTqE69evl1wvk8ng7Oxc6SR5RESESRsf3r73dyA1b9yo3Pfv3/iQf5dkLgwGK5KXl4eAgAB8+eWXWLduHYKDgyu9vnfv3qV+WNy/WIpMJ5PJ0KJFC7Ro0QK9evUq95qCggJkZGSUO8+RlJSEjIwMFBYWllxvb2cH/+GDK934UAiBBeu2YJBXN3h26lDuNdz4kGoDg8FK/Pnnnxg3bhx++eUXfPnll3jqqaeqfA+GQu1xcHCARqOBRqMp932j0Yjr16+XBEbA1KkP3fhw3qoNOJP+B45vWV3pdV4aD+yO2Vnt2okexGCwAmfPnsWYMWNQWFiIH374gY8MrJBcLoeLiwtcXFzw2GOPYfLkyZVufBi8eiMOxCfi2KZVaNe6VaX3Lt74kCNCMhd+F1m4o0ePYuDAgWjcuDF++uknhoINqGzjQyEE5q3agP1xP+Lo+g/Qsa3zQ+93+14uVCoVQ4HMht9JFmz79u0YNWoUfHx8EB8fj/bt2z/8g8gqVLTx4dxVGxB1OAZR77yORg0c8edf2fjzr2zkF+gqvFeK9gK6e3rWZrlUzzAYLJDRaMSSJUswc+ZMzJw5EwcOHEDjxo2lLovMyGfAABw5mQK93lDq9c37DuL2vVz4zl2Itk9NKfmz++ixcu+j1xtwNCkF/X186qJsqie4jsHCFBQUYPr06fjss88QEhKCBQsW1PsdR22RpW58SDaPC9yszc2bN/H0008jOTkZkZGRePbZZ6UuiWqRn68vLmnPIWXXRqvZ+JCsnknBwEdJFuLcuXPo378/0tPTERcXx1CoB8K2bkVmdg6CQkJLbclhCqPRiKCQUGRm5yBs69ZaqpDqKwaDBTh+/Dh8fHxgb2+PxMRE9OvXT+qSqA6o1WqEh4cj8nAMApetRm5+gUkfl5tfgMBlqxF5OAbh4eH1ejsTqh0MBolFRUVh2LBh8PLywo8//siTueoZf39/REZGYs+xE/CaFoR9cfFlJqSL6fUG7IuLR4+Al7Dn2Amr3A2XrAPnGCQihMDy5cvxn//8B9OnT8eWLVts9nQyeriqbHzYwNERP/38MzzZokpVx8lnS1VYWIg5c+Zg586dWLZsGd544w12HhEAlNr4MDUtrdTGh/19fDB06FD84x//wH//+18sXrxY6nLJ+jAYLFFOTg6eeeYZnDhxAhERETx9iypV3jYX8+fPx7Zt26DVauHs/PCV0UT3YTBYmgsXLmDMmDHIysrCl19+iUGDBkldElmh7OxsqNVqTJw4EWFhYVKXQ9aF7aqWJDExEf3794der0diYiJDgaqtefPmePvttxEeHo4zZ85IXQ7ZII4Y6sCePXswbdo0eHt744svvkDLli2lLomsXFFRETw9PeHm5obvvvuOc1RkKo4YpCaEwMqVK/GPf/wDEyZMwJEjRxgKZBZ2dnZYtWoVjhw5gq+//lrqcsjGcMRQS4qKijBv3jyEhYXhjTfewH//+19ui0xmJYTAsGHDcPXqVaSmpsLOzk7qksjyccQglTt37uCpp57C9u3bsX37dixfvpyhQGYnk8mwZs0anDt3Dlu2bJG6HLIhHDGYWUZGBsaMGYPLly9j37598PPzk7oksnGzZs3C/v37kZ6ejmbNmkldDlk2jhjqWlJSEvr164e7d+/ixIkTDAWqE8uXL4dOp8Py5culLoVsBIPBTA4cOIAnnngC7du3R2JiIh599FGpS6J6wtnZGYsXL0ZoaCi0Wq3U5ZAN4KMkMwgNDcUrr7yC8ePHY9euXWjQoOJD3olqQ35+Prp06YI+ffpg3759UpdDlouPkmqbwWDAyy+/jH/961949dVX8fnnnzMUSBKOjo54//33sX//fsTFxUldDlk5jhiq6d69e5gyZQoOHTqE9evX45///KfUJVE9J4SAj48PCgsLcfLkSXbCUXk4Yqgt165dw+DBgxEbG4uDBw8yFMgiFLevnjp1Ch9//LHU5ZAV44ihilJTUzFmzBgYjUYcOnQIXl5eUpdEVMrkyZNx/PhxaLVaODk5SV0OWRaOGMzt8OHDGDhwIFq2bImffvqJoUAW6YMPPkB2djZWrlwpdSlkpRgMJgoLC8OYMWPwxBNP4IcffoCrq6vUJRGVq0OHDnjllVcQEhKCK1euSF0OWSEGw0MYjUa8/vrrePHFF/HSSy/hiy++QMOGDaUui6hSS5YsQcOGDfHGG29IXQpZIQZDJfLz8zFp0iSsXLkSa9euRWhoKJRKpdRlET1U48aNsWzZMnz88cc4efKk1OWQleHkcwWysrIwbtw4pKamIjo6Gk8//bTUJRFViV6vR69evdCsWTMcO3aMZzYQwMnn6vv111/Rv39/XLp0CceOHWMokFVSKpVYvXo1jh8/ztXQVCUcMTwgNjYWzzzzDFxdXXHo0CG4u7tLXRJRjYwZMwa//fYbfvnlF6hUKqnLIWlxxFBVO3fuxMiRI/HYY4/hxx9/ZCiQTVi1ahUuXbqE0NBQqUshK8FgwN9bCbz11luYPn06XnjhBRw6dAhNmjSRuiwis3jkkUfw0ksvYdmyZbhx44bU5ZAVsNlHSUaj0aS9YnQ6HWbOnImoqCisWLECr7/+OifpyObcvHkTarUaU6dOxYYNG6Quh6RTvx4lJScnIzg4GH28veHg4ACFQgEHBwf08fZGcHAwkpOTy3zMX3/9heHDh2PPnj3YvXs3Fi1axFAgm9SyZUssXboUW7ZswS+//CJ1OWThrH7EkJ6ejjmzZyM2Lg6urVthWB8veGk6obFTA9zJzUOK9jyOnEzB1awb8B0yBGFbt0KtViM9PR2jR49GTk4OvvrqK/j4+Ej9pRDVKp1Oh0cffRRdu3bFoUOHpC6HpGHSb75WHQzR0dGYNWsWXJo3w8p5MzF2UH8olYoy1+n1BhyIT8Rr68ORmZ2DhQsXYv369WjZsiUOHTqETp06SVA9Ud3bu3cvJk6ciMOHD2PEiBFSl0N1z7aDITo6GgEBAQgY6YeNC4Ph5Ojw0I/JzS/AP0NCEfntUXTt2hU//vgjmjdvXgfVElkGIQQGDx6M7OxsnD59miv56x/bnWPQarWYNWsWAkb6IWLpApNCAQCcHB2wY+kCBIzyw6VLF5GdnV3LlRJZluIzG86ePYvw8HCpyyELZZUjBj9fX1xOP4fTH28sFQpXs25i0cZwfJNwEvm6QnR2c8W2JfPh3VVT6uNz8wvgNS0I7prOiImNrevyiST3wgsv4JtvvkF6ejoaN24sdTlUd2xzxJCUlITYuDiEzJ1ZKhRy7tzFoBdfhZ1Sia/XLMfZT7ZgVfBsNG1Y9qASJ0cHhMybgdi4uHK7lYhs3bvvvot79+7hvffek7oUskBWFww7duxAuzatMXZQ/1KvfxD5Odq3aYXtby5A325d0MHFGUMf64VO7dqWe59xg3zg2roVIiIi6qJsIovSrl07LFy4EGvXrsUff/whdTlkYawuGBJOnMBQ7x5luo8OHE+Ed9fOmLRkOdqMnozez8/F1i+/qfA+SqUCQ729kJiQUNslE1mk1157DS1btsSiRYukLoUsjNUFQ9rZs/DSlG0vvXAtE5v3H4S6vSu+XfsuXpwwGi+v2YSPvz5S4b28NB5ITUurzXKJLJaTkxPee+89fPbZZzhx4oTU5ZAFsapgMBqN0Ol0aOzUoJz3BHp3VuO9fwaiVxc1XpwwBrOeHoXN+w9WeL8mDZ2g0+lgNBprs2wiizVt2jT07t0b8+fP538HVMKqgkEul0OlUuFObl6Z91xaNscjHd1KvfZIBzdc/rPiTcNu38uFSqUyaU8lIlskl8uxZs0a/Pzzz/j000+lLocshNX9RPTs1g0p2vNlXh/Y/VGcu1z64PNzl6/C3bl1hfdK0V5Ad09Ps9dIZE0GDx6MCRMmYNGiRcjPz5e6HLIAVhcMPgMG4MjJFOj1hlKvv/LcBCSm/Yb3dnyK9IxriD4ci61ffo2giWPLvY9eb8DRpBT05x5JRAgJCcGff/6JNWvWSF0KWQCrW+CWnJwMb29v7F2xFBOGDCz13sH4n7BkUwS0V66io4sz5vs/g9lPP1nuffbFxWPi4uVISkpC796966J0Iou2YMECbNmyBenp6XB2dpa6HKodtrtXkp+vLy5pzyFl10aTt8O4H1c+E5V169YtqNVqjB8/Htu2bZO6HKodtrnyGQDCtm5FZnYOgkJCq9xJYTQaERQSiszsHIRt3VpLFRJZn6ZNm+Ltt9/G9u3bkZKSInU5JCGrHDEAwCeffIKpU6dWeXfVoJBQRB6OQVRUFPz9/eugUiLrUVRUhB49eqBt27Y4cuQID66yPbY7YgAAf39/REZGYs+xE/CaFoR9cfFlJqSL6fUG7IuLh9e0IOw5doKhQFQBOzs7rFq1CjExMTh4sOI1QGTbrHbEUOzBE9yGenvBS+OBJg2dcPteLlK0F3A06e8T3Px8fbElLAxqtVrqsokslhACI0aMwOXLl5Gamgp7e3upSyLzsd3J5/IkJycjIiICiQkJSE1Lg06ng0qlQndPT/T38UFgYCC7j4hMlJqaip49e2Lt2rX417/+JXU5ZD71KxgeZDQauaKZqAbmzJmDPXv2ID09nScd2g7bnmN4GIYCUc0sW7YMRUVFWLZsmdSlUB3jT08iKlebNm2wZMkSrF+/HufOnZO6HKpDNvsoiYhqrqCgAF27dkXPnj3xxRdfSF0O1Vz9fpRERDXn4OCA999/H19++SViuUtAvcERAxFVSgiBAQMGoKCgACdPnoRCoXj4B5Gl4oiBiGpOJpNh7dq1OH36ND7++GOpy6E6wBEDEZlkypQpiI2NhVarRcOGDaUuh6qHIwYiMp8VK1YgJycHISEhUpdCtYzBQEQmcXd3x6uvvopVq1YhIyND6nKoFvFREhGZ7O7du1Cr1RgxYgR27doldTlUdXyURETm1ahRIyxfvhyRkZH4+eefpS6HaglHDERUJQaDAb169ULjxo1x/PjxMmc2cJ8yi8YRAxGZn0KhwJo1a/Djjz9iz549SE5ORnBwMPp4e8PBwQEKhQIODg7o4+2N4OBgJCcnS10yVRFHDERULX5+fvj5p5+Qm5cH19atMKyPF7w0ndDYqQHu5OYhRXseR07+fRaK75AhCNu6lWehSM+kEYOytqsgItsTHR2NhIQEtG7SGB8vXYCxg/pDqSy7IlqvN+BAfCJeWx+OHj16IDw8nKcnWgGOGIioSqKjoxEQEFDt89YjIyMxZcqUOqiUylG/D+ohIvPTarXw8vLCxMEDELF0Qckkc8cJz+PSn1llrv/nM09hw2vzAPw9KR24bDX2HDuBM2fO8LGSNBgMRGRefr6+uJx+Dqc/3lhqpHAj5xYMRmPJv6edv4gRLy9BzIYPMKS3V8nrufkF8JoWBHdNZ8Rwt1YpsCuJiMwnKSkJsXFxCJk7s8zjo1bNmsK5RfOSPwd//BmdXF0wuFePUtc5OTogZN4MxMbFsVvJgjEYiMgkO3bsQLs2rTF2UP9KryssKkLU4RgEPjWyzBoHABg3yAeurVshIiKitkqlGmIwEJFJEk6cwFDvHuV2H93vi2MJuHXvHqaPGV7u+0qlAkO9vZCYkFAbZZIZMBiIyCRpZ8/CS9PpoddtP/gtnuz/GNq2alHhNV4aD6SmpZmzPDIjBgMRPZTRaIROp0NjpwaVXncp8zqO/O80Zo4bVel1TRo6QafTwXjfhDVZDgYDET2UXC6HSqXCndy8Sq+LOPQdWjdrgjED+lZ63e17uVCpVNxTyULxb4WITOLZrRtStOcrfN9oNGLHoe/x/OjhD52HSNFeQHdPT3OXSGbCYCAik/gMGIAjJ1Og1xvKff/I/07h8p9ZmPHUiErvo9cbcDQpBf19fGqjTDIDBgMRmSQwMBBXs27gQHxiue+P6OcNY8K36OzWrtL7fBWfgKtZNxAYGFgbZZIZcOUzEZnMz9cXl7TnkLJro0l7JD2IK58lx5XPRGReYVu3IjM7B0EhoVXuKDIajQgKCUVmdg7Ctm6tpQrJHKoyYiAigkwm8wcQFTBqKDYtDJaZurvqP0NCReS3RwFgqhDik9quk6qPwUBEVSaTyabI5bLt7Vu3Uqx+eY5y3CCfCs9j+Co+AQs+CtNnZN0wGI0ikKFg+RgMRFQtMplMrVQowvUGwxNtmjfTj+znrfTSeKBJQyfcvpeLFO0FHP4pSX89O0epVCiO6Q2GWUKIdKnrpodjMBBRjchkst4AAu2UikEGg7GbUQg7uUxWpJDLzxYZDPEAIoQQ3ErVijAYiMisZDKZXAjBvS6sGIOBiIhKYbsqERGVwmAgIqJSGAxERFQKg4GIiEr5f4v84JK6htzjAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics object consisting of 21 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g=Graph({0:[1,5],1:[2,6],2:[3,7],3:[4,8],4:[0,9]})\n",
    "show(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "5c4adef9-a874-459d-b840-982e2af44831",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 3]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "up_low_ZIR(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "606cecdb-db6f-45b0-9dc9-60764de3db20",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Example 7.4: Verify that zir(G)=4 for the graph G in Figure 7.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f91f87d5-8898-4bc3-873e-aba7a02f4873",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 26 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g=graphs.CompleteBipartiteGraph(4,4)\n",
    "g.add_edges([(0,1),(5,4)])\n",
    "g.delete_edge([3,7])\n",
    "show(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "88520b09-450a-4004-8f85-338f770d5684",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[5, 4]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "up_low_ZIR(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "34b90e20-266b-432f-8703-ad0075493442",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.treewidth()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "71bf18b2-4c18-4a94-a694-54159bda2d7a",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/lib/python3/dist-packages/sage/misc/remote_file.py:46: DeprecationWarning: ssl.SSLContext() without protocol argument is deprecated.\n",
      "  content = urlopen(req, timeout=1, context=SSLContext())\n",
      "/usr/lib/python3/dist-packages/sage/misc/remote_file.py:46: DeprecationWarning: ssl.PROTOCOL_TLS is deprecated\n",
      "  content = urlopen(req, timeout=1, context=SSLContext())\n",
      "Compiling /home/jupyter-hogben@iastate.edu-c55cc/.sage/temp/sagemath2/464433/tmp_w27tzsmg.pyx...\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "xrange test failed: define xrange = range\n",
      "Loading Zq_c.pyx...\n",
      "Loading Zq.py...\n",
      "Loading zero_forcing_64.pyx...\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Compiling /home/jupyter-hogben@iastate.edu-c55cc/.sage/temp/sagemath2/464433/tmp_ph4ft_98.pyx...\n",
      "Compiling /home/jupyter-hogben@iastate.edu-c55cc/.sage/temp/sagemath2/464433/tmp_90gk2z3z.pyx...\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Loading zero_forcing_wavefront.pyx...\n",
      "Loading minrank.py...\n",
      "Loading inertia.py...\n"
     ]
    }
   ],
   "source": [
    "# Cell F-ZIR1\n",
    "# load main minimum rank/maximum nullity and zero forcing functions\n",
    "# developed by S. Butler, L. DeLoss, J. Grout, H.T. Hall, J. LaGrange, J.C.-H. Lin,T. McKay, J. Smith, G. Tims\n",
    "# updated and maintained by Jephian C.-H. Lin \n",
    "load(\"https://raw.githubusercontent.com/jephianlin/minrank_aux/master/load_all.py\")\n",
    "load_all(mr_JG=True, minrank_aux=False, timeout=5, load_func=\"loadurl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "3d874706-c5e9-47d2-b83c-c65d70f151d5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Cell F-ZIR2\n",
    "# Useful tools\n",
    "\n",
    "\n",
    "# function: is_fort(G,F)\n",
    "# input: graph G and set F of vertices \n",
    "# output: True if F is a fort, False if F is not a fort\n",
    "\n",
    "def is_fort(G,F):\n",
    "    # initialize flag isFort\n",
    "    if len(F) > 0: \n",
    "        isFort = True\n",
    "    else:\n",
    "        isFort = False\n",
    "    F = set(F)\n",
    "    V = set(G.vertices())\n",
    "    Z = V - F\n",
    "    # check if a vertex in the complement of F is adjacent to exactly 1 vertex in F\n",
    "    for v in Z:\n",
    "        if len(set(G.neighbors(v)) & F) == 1:\n",
    "            isFort = False\n",
    "            break\n",
    "    return isFort\n",
    "\n",
    "\n",
    "# function: kforts(G,k)\n",
    "# input: a graph G and a positive integer k \n",
    "# output: a list of all forts of G of size k\n",
    "\n",
    "def kforts(G,k):\n",
    "    V = G.vertices()\n",
    "    S = Subsets(V,k)    \n",
    "    forts = []\n",
    "    for f in S:\n",
    "        if is_fort(G,f):\n",
    "            forts.append(set(f))\n",
    "    return forts\n",
    "\n",
    "\n",
    "# function: all_forts(G)\n",
    "# input: a graph G\n",
    "# output: a list of all forts of G\n",
    "\n",
    "def all_forts(G):\n",
    "    V = set(G.vertices())\n",
    "    S = Subsets(V)    \n",
    "    forts = []\n",
    "    for f in S:\n",
    "        if is_fort(G,f):\n",
    "            forts.append(set(f))\n",
    "    return forts\n",
    "    \n",
    "\n",
    "# function: is_ZIRset(G,T)\n",
    "# input: graph G and set of vertices T with at least 1 element\n",
    "# output: True if T is a ZIR set and False otherwise\n",
    "\n",
    "def is_ZIRset(G,T):\n",
    "    T = set(T)\n",
    "    S = set([])\n",
    "    ZIR = False\n",
    "    forts = all_forts(G)\n",
    "    for F in forts:\n",
    "        F_int = F.intersection(T)\n",
    "        if len(F_int) == 1:\n",
    "            S = S.union(F_int)\n",
    "        if len(S) == len(T):\n",
    "            ZIR = True\n",
    "            break\n",
    "    return ZIR\n",
    "\n",
    "\n",
    "# function: ZIRset_cert(G,T)\n",
    "# input: graph G and set T\n",
    "# output: False if T is not a ZIR set, otherwise private forts\n",
    "\n",
    "def ZIRset_cert(G,T):\n",
    "    T = set(T)\n",
    "    S = set([])\n",
    "    ZIR = False\n",
    "    cert = []\n",
    "    forts = all_forts(G)\n",
    "    for F in forts:\n",
    "        F_int = set(F).intersection(T)\n",
    "        if len(F_int) == 1:\n",
    "            cert.append(F)\n",
    "            S = S.union(F_int)\n",
    "        if len(S) == len(T):\n",
    "            ZIR = True\n",
    "            break\n",
    "    return cert if ZIR == True else ZIR\n",
    "    \n",
    "\n",
    "# function: mult_ZIR_cert(G,T,forts)\n",
    "# input: graph G, set T, and list of all forts\n",
    "# output: False if T is not a ZIR set, otherwise private forts\n",
    "\n",
    "def mult_ZIRset_cert(G,T,forts):\n",
    "    T = set(T)\n",
    "    S = set([])\n",
    "    ZIR = False\n",
    "    cert = []\n",
    "    for F in forts:\n",
    "        F_int = set(F).intersection(T)\n",
    "        if len(F_int) == 1:\n",
    "            cert.append(F)\n",
    "            S = S.union(F_int)\n",
    "        if len(S) == len(T):\n",
    "            ZIR = True\n",
    "            break\n",
    "    return cert if ZIR == True else ZIR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "51943646-038e-4263-a77f-05ea9ca82346",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Cell F-ZIR3\n",
    "# Function for finding ZIR sets\n",
    "\n",
    "# Possible improvements: Carry fort data with vertices - would reduce checking if ZIR set\n",
    "#                        Incorporating bounds on lower zir number (min degree)\n",
    "\n",
    "\n",
    "\n",
    "# function: is_ZIRset_mult(G,T,forts)\n",
    "# input: graph G, set of vertices T with at least 1 element, list of all forts\n",
    "# output: True if T is a ZIR set and False otherwise\n",
    "\n",
    "def is_ZIRset_mult(G,T,forts):\n",
    "    T = set(T)\n",
    "    S = set([])\n",
    "    ZIR = False\n",
    "    for F in forts:\n",
    "        F_int = F.intersection(T)\n",
    "        if len(F_int) == 1:\n",
    "            S = S.union(F_int)\n",
    "        if len(S) == len(T):\n",
    "            ZIR = True\n",
    "            break\n",
    "    return ZIR\n",
    "\n",
    "\n",
    "# function: extend(G,L,k)\n",
    "# input: graph G, list L, and integer k\n",
    "# output: add vertices to L to increase size of ZIR set\n",
    "\n",
    "def extend(G,L,forts):\n",
    "    n = G.order()\n",
    "    k = L[-1:][0]    # last element of L\n",
    "    new_L = list(L)\n",
    "    for i in range(k+1,n+1):    # add each vertex greater than k to L in consecutive order if resulting list is ZIR set \n",
    "        if is_ZIRset_mult(G,new_L + [i],forts):\n",
    "            new_L.append(i)\n",
    "    return new_L\n",
    "\n",
    "\n",
    "# function: update(L,ord)\n",
    "# input: list L, positive integer ord\n",
    "# output: updates L to the next list that needs to be checked\n",
    "\n",
    "# function: update(L,ord)\n",
    "# input: list L, positive integer ord\n",
    "# output: updates L to the next list that needs to be checked\n",
    "\n",
    "def update(L,ord):\n",
    "    n = len(L)\n",
    "    flag = False\n",
    "    if ord - L[-1:][0] > 1:                               # check if L contains the largest vertex and proceed if not\n",
    "        new_L = [x for x in L if x < L[-1:][0]]           # remove largest vertex v from L\n",
    "        new_L.append(L[-1:][0] + 1)                       # add vertex v+1 to L\n",
    "        flag = True\n",
    "    else:                                                 # runs the following if L contains largest vertex\n",
    "        for s,t in zip(reversed(L),reversed(L[:n-1])):    # zip L and copy of L with last element removed\n",
    "            if s - t > 1:                                 # check for non consecutive vertices\n",
    "                flag = True\n",
    "                new_L = [x for x in L if x < t]           # once non consecutive pair (t,s) is found (t < w), remove t and all vertices after t\n",
    "                new_L.append(t+1)                         # append t+1\n",
    "                break\n",
    "    return new_L if flag == True else False\n",
    "\n",
    "\n",
    "# function: max_subsets(sets)\n",
    "# input: list of lists\n",
    "# output: removes subsets\n",
    "\n",
    "def max_subsets(sets):\n",
    "    sets = reversed(sorted(map(set, sets), key=len))\n",
    "    maximal_subsets = []\n",
    "    for s in sets:\n",
    "        if not any(s.issubset(maximal_subset) for maximal_subset in maximal_subsets):\n",
    "            maximal_subsets.append(s)\n",
    "    return maximal_subsets\n",
    "\n",
    "\n",
    "# function: ZIRsets(G)\n",
    "# input: graph G\n",
    "# output: all maximal ZIR sets\n",
    "\n",
    "def ZIRsets(G):\n",
    "    ord = G.order()\n",
    "    forts = all_forts(G)\n",
    "    L = [0]\n",
    "    zsets = []    # initialize all maximal ZIR sets\n",
    "    flag1 = True\n",
    "    while flag1 == True:\n",
    "        L = extend(G,L,forts)    # extend L to obtain maximal ZIR set\n",
    "        zsets.append(L)\n",
    "        flag2 = True\n",
    "        while flag2 == True:    # update L to next ZIR set (still needs to be extended) and check if done\n",
    "            L = update(L,ord)\n",
    "            if L == False:\n",
    "                flag1 = False\n",
    "                flag2 = False\n",
    "            elif is_ZIRset_mult(G,L,forts):\n",
    "                flag2 = False\n",
    "    zsets = max_subsets(zsets)     # remove ZIR sets that are not maximal\n",
    "    return zsets\n",
    "\n",
    "\n",
    "# function: up_low_ZIR(G)\n",
    "# input: graph G\n",
    "# output: upper and lower ZIR number\n",
    "\n",
    "def up_low_ZIR(G):\n",
    "    zsets = ZIRsets(G)\n",
    "    return [len(zsets[0]),len(zsets[-1])]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d0728432-fff0-4acb-b7cf-1a213563d3af",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Cell F-ZIR4 Zero forcing sets \n",
    "#\n",
    "# Zero forcing number Z(G)\n",
    "def Z(g):\n",
    "    z = len(zero_forcing_set_bruteforce(g))\n",
    "    return z\n",
    "\n",
    "# All zero forcing sets of G of size k\n",
    "# input: a graph G and a positive integer k \n",
    "# output: a list of all zero forcing sets of G of size k\n",
    "def ZFsets(G,k):\n",
    "    ord=G.order()\n",
    "    V = G.vertices()\n",
    "    AllSubs = set(Subsets(V,k))\n",
    "    ZF=[]\n",
    "    for s in AllSubs:\n",
    "        A = zerosgame(G,s)\n",
    "        if len(A) == ord:\n",
    "            ZF.append(s)\n",
    "    return ZF\n",
    "\n",
    "def all_ZFsets(G,k): \n",
    "    S = []\n",
    "    for i in [Z(G)..k]:\n",
    "        ZFS_size_i = ZFsets(G,i)\n",
    "        S = S + ZFS_size_i\n",
    "    return S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2228d372-ce06-47ef-8dd7-40773433d3fd",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Cell F-ZIR5 \n",
    "#\n",
    "# Input: list of lists/sets/tuples\n",
    "# Output: list without super-lists/sets/tuples\n",
    "   \n",
    "def get_minimal_subsets(sets):\n",
    "    out = [x for x in sets if not any(set(x).issuperset(set(y)) and len(x)>len(y) for y in sets)]\n",
    "    return out\n",
    "\n",
    "\n",
    "# Input: graph G\n",
    "# Output: all minimal zero forcing sets\n",
    "\n",
    "def ZF_min_sets(G):\n",
    "    ord = G.order()\n",
    "    ZFS = all_ZFsets(G,ord)\n",
    "    mZFS = get_minimal_subsets(ZFS)\n",
    "    return mZFS\n",
    "\n",
    "\n",
    "# Input: graph G\n",
    "# Output: the zero forcing number and upper minimal zero forcing number\n",
    "\n",
    "def zbar(G):\n",
    "    min_list = ZF_min_sets(G)\n",
    "    Zb = max(len(x) for x in min_list)\n",
    "    return Zb\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1c306c2b-00af-462f-90a3-2f2b5b7b68a6",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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