On the Pairwise Compatibility Property of Some Superclasses of Threshold Graphs

Abstract

A graph G is called a pairwise compatibility graph (PCG) if there exists a positive edge weighted tree T and two non-negative real numbers d min and d max such that each leaf lu of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if d min ≤ dT (lu, lv) ≤ d max , where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we study the relations between the pairwise compatibility property and superclasses of threshold graphs, i.e., graphs where the neighborhoods of any couple of nodes either coincide or are included one into the other. Namely, we prove that some of these superclasses belong to the PCG class. Moreover, we tackle the problem of characterizing the class of graphs that are PCGs of a star, deducing that also these graphs are a generalization of threshold graphs.

Pairwise Compatibility Graphs: A Survey

“…This result was extended to the following larger subclass of split matrogenic graphs [21]. In fact, it seems that the order of appearance of a split matching or an split antimatching in the composition of a split matrogenic graph is somehow strictly related to the pairwise compatibility property.…”

“…In [21] it is proved that if the split matrogenic graph is composed using only split matching graphs or only split anti-matching graphs, then it belongs to PCG.…”

A survey on pairwise compatibility graphs

“…Every graph with at most seven vertices [7], Cycles, Single chord cycles [35]; Ladder graphs, Block-cycle graphs, Triangle-free outerplanar graphs [30]; Graphs with Dilworth number at most two [9]; A restricted subclass of split matrogenic graphs [11]; A restricted subclass of bipartite graphs, Tree power graphs [34] LPGs Trees [20], Interval graphs, Ptolemaic graphs [3]; Threshold graphs, Split matching graphs [8], A superclass of directed rooted graphs [4] mLPGs Threshold tolerance graphs [12], Split antimatching graphs [8] 2-interval PCGs Wheel graphs, A restricted subclass of series-parallel graphs [1] belonging to the clique forms a perfect matching and an antimatching respectively [10]. A graph is an intersection graph if the vertices corresponds to a family of sets and there is an edge between two vertices if the corresponding sets have non-empty intersection.…”

Characterizing Star-PCGs

“…This graph class is known as min leaf power graph (mLPG) [6], which is the complement of LPG. Several other known graph classes have been shown to be subclasses of PCG, e.g., disjoint union of cliques [2], forests [11], chordless cycles and single chord cycles [19], tree power graphs [18], threshold graphs [6], triangle-free outerplanar 3-graphs [16], some particular subclasses of split matrogenic graphs [6], Dilworth 2 graphs [5], the complement of a forest [11] and so on. It is also known that a PCG with a witness tree being a caterpillar also allows a witness tree being a centipede [4].…”

“…Although stars are trees with a rather simple topology, star-PCG recognition is not easy at all. It is known that threshold graphs are star-PCGs (even in star-LPG and star-mLPG) and the class of star-PCGs is nearly the class of threethreshold graphs, a graph class extended from the threshold graphs [6]. However, no complete characterization of star-PCGs and no polynomial-time recognition of star-PCGs are known.…”