# P4-structure and Its Relatives

The -structure of a graph is the 4-uniform hypergraph whose vertex-set is the vertex-set of and whose hyperedges are the subsets of that induce 's (chordless paths on four vertices) in . The graph property of being Berge can be formulated directly in terms of -structure: a graph is Berge if and only if its -structure contains no induced odd ring, meaning a 4-uniform hypergraph with vertices where is odd and at least five, and with the hyperedges where the subscripts are taken modulo . (The if'' part is trivial and the only if'' part is easy, even though a little tedious; see [ MR 86j:05119] for a sketch of the argument.)

Can the Chudnovsky-Robertson-Seymour-Thomas decomposition theorem for Berge graphs be reformulated directly in terms of -structure?

The five classes of basic graphs featured in the theorem lend themselves nicely to such reformulations: there are classes C of 4-uniform hypergraphs such that

• the -structure of every basic graph belongs to C,
• no 4-uniform hypergraph in C contains an induced odd ring,
• membership in C can be tested in polynomial time.
One such class is defined in terms of a certain directed graph, , associated with every 4-uniform hypergraph : C consists of all 4-uniform hypergraphs such that
• contains no induced ring with five vertices and
• all strongly connected components of are bipartite.
The vertices of are all the ordered 6-tuples of distinct vertices of such that the sub-hypergraph of induced by the set consists of the three hyperedges there is a directed edge from vertex of to vertex of if and only if Trivially, membership in C can be tested in polynomial time; trivially, no 4-uniform hypergraph in C contains an induced odd ring; a proof that the -structure of every basic graph belongs to C is easy, even though a little tedious ([here] is a sketch of the argument).

The four kinds of structural faults featured in the decomposition theorem suggest the following three problems.

Problem 1: Find a class of 4-uniform hypergraphs such that

• the -structure of every graph with a 2-join belongs to ,
• no odd ring belongs to ,
• belongs to NP.

Problem 2: Find a class of 4-uniform hypergraphs such that

• the -structure of every graph with an M-join belongs to ,
• no odd ring belongs to ,
• belongs to NP.

Problem 3: Find a class of 4-uniform hypergraphs such that

• the -structure of every graph with a balanced skew partition belongs to ,
• no odd ring belongs to ,
• belongs to NP.

Chính Hoàng introduced additional graph functions that are invariant under complementation and determine whether or not a graph is Berge: these are the co-paw-structure ([ MR 2001a:05065]), the co- -structure (Discrete Math. 252 (2002), 141-159), and the co- -structure (to appear in SIAM J. Discrete Math.). Can the decomposition theorem be reformulated directly in terms of one of Hoàng's invariants?

Contributed by Vašek Chvátal

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