On local structures of cubicity 2 graphs

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation

A New Graph Parameter to Measure Linearity

International audienceSince its introduction to recognize chordal graphs by Rose, Tarjan, and Lueker, Lexicographic Breadth First Search (LexBFS) has been used to come up with simple, often linear time, algorithms on various classes of graphs. These algorithms are usually multi-sweep algorithms; that is they compute LexBFS orderings σ1,…,σkσ1,…,σk , where σiσi is used to break ties for σi+1σi+1 . Since the number of LexBFS orderings for a graph is finite, this infinite sequence {σi}{σi} must have a loop, i.e. a multi-sweep algorithm will loop back to compute σjσj , for some j. We study this new graph invariant, LexCycle(G), defined as the maximum length of a cycle of vertex orderings obtained via a sequence of LexBFS ++ . In this work, we focus on graph classes with small LexCycle. We give evidence that a small LexCycle often leads to linear structure that has been exploited algorithmically on a number of graph classes. In particular, we show that for proper interval, interval, co-bipartite, domino-free cocomparability graphs, as well as trees, there exists two orderings σσ and ττ such that σ=LexBFS+(τ)σ=LexBFS+(τ) and τ=LexBFS+(σ)τ=LexBFS+(σ) . One of the consequences of these results is the simplest algorithm to compute a transitive orientation for these graph classes. It was conjectured by Stacho [2015] that LexCycle is at most the asteroidal number of the graph class, we disprove this conjecture by giving a construction for which LexCycle(G)>an(G)LexCycle(G)>an(G) , the asteroidal number of G

A Benders Approach to the Minimum Chordal Completion Problem

This paper introduces an integer programming approach to the minimum chordal completion problem. This combinatorial optimization problem, although simple to pose, presents con-siderable computational difficulties and has been tackled mostly by heuristics. In this paper, an integer programming approach based on Benders decomposition is presented. Computa-tional results show that the improvement in solution times over a simple branch-and-bound algorithm is substantial. The results also indicate that the value of the solutions obtained by a state-of-the-art heuristic can be in some cases significantly far away from the previously unknown optimal solutions obtained via the Benders approach