Characterizing ‐perfect line graphs

Abstract

The aim of this paper is to study the Lovász-Schrijver PSD operator N + applied to the edge relaxation of the stable set polytope of a graph. We are particularly interested in the problem of characterizing graphs for which N + generates the stable set polytope in one step, called N + -perfect graphs. It is conjectured that the only N + -perfect graphs are those whose stable set polytope is described by inequalities with near-bipartite support. So far, this conjecture has been proved for near-perfect graphs, fs-perfect graphs, and webs. Here, we verify it for line graphs, by proving that in an N + -perfect line graph the only facet-defining subgraphs are cliques and odd holes.

Lovász-Schrijver PSD-Operator on Claw-Free Graphs

“…Together with the result from [12] (presented in Theorem 5), we directly conclude that the LS + -Perfect Graph Conjecture holds for semi-line graphs.…”

“…, H k = H where H 0 is an odd hole and H i is obtained from H i−1 by adding an odd path (ear) between distinct nodes of H i−1 . In [12], it is shown that the line graph L(H 1 ) is a node stretching of G LT or G EMN and, thus, LS + -imperfect by [22].…”

“…In [1], the authors look for a characterization of LS + -perfect graphs similar to the characterization (1) for perfect graphs: they intend to find an appropriate polyhedral relaxation P (G) of STAB(G) such that G is LS + -perfect if and only if STAB(G) = P (G). A conjecture has been recently proposed in [2], which can be equivalently reformulated as follows [12]:…”

Lovász-Schrijver PSD-Operator on Some Graph Classes Defined by Clique Cutsets

“…Conjecture 1 has already been verified for several graph classes: fs-perfect graphs [1] (where the only facet-defining subgraphs are cliques and the graph itself), webs [12] (the complements W k n = A k n of antiwebs), line graphs [13] (obtained by turning adjacent edges of a root graph into adjacent nodes of the line graph), and claw-free graphs [2]; the latter result includes graphs G with stability number α(G) at most 2.…”

“…Graphs G with STAB(G) = LS + (G) are called LS + -perfect, and all other graphs LS + -imperfect. A conjecture has been proposed in [1], which can be equivalently reformulated as follows, as noted in [13]:…”