Asteroidal quadruples in non rooted path graphs

Abstract

International audienceA directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted path graphs. For this purpose, we are studying in this paper directed path graphs that are non rooted path graphs. We prove that such graphs always contain an asteroidal quadruple

On Models of Directed Path Graphs Non Rooted Directed Path Graphs

“…As was proved in [6], this kind of connections are not the only that appears in DV minimally non RDV graphs.…”

“…The first equality is a consequence of [4], the second equality has been proved independently in [3] and [7]. Surprisingly for RDV graph the equality does not hold [6].…”

“…An asteroidal quadruple in a graph G is a set of four non adjacent vertices such that any three of them is an asteroidal triple. In its original form, this conjecture is incomplete as was showed in [6]. Gutierrez, Lévêque and Tondato [6] proved also that every directed path graph that is not rooted path graph has an asteroidal quadruple.…”

“…In its original form, this conjecture is incomplete as was showed in [6]. Gutierrez, Lévêque and Tondato [6] proved also that every directed path graph that is not rooted path graph has an asteroidal quadruple. But clearly, it is not a characterization.…”

Special asteroidal quadruple on directed path graph non rooted path graph

“…They also introduce a related notion of asteroidal quadruple and conjecture a characterization of rooted path graphs [1]. In its original form this conjecture is not complete, still in leafage four, as was showed in [3]. But, as suggested by the conjecture, a characterization by forbidding particular types of asteroidal quadruples may hold.…”