Pairwise compatibility graphs

Abstract

Given an edge weighted tree T and two non-negative real numbers d min and d max , a pairwise compatibility graph (PCG) of T is a graph G = (V , E), where each vertex u ∈ V corresponds to a leaf u of T and an edge (u, v) In this paper we give some properties of these graphs. We establish a relationship between pairwise compatibility graphs and chordal graphs. We show that all chordless cycles and single chord cycles are pairwise compatibility graphs. We also provide a linear-time algorithm for constructing trees that can generate graphs having cycles as their maximal biconnected subgraphs as PCGs. The techniques that we used to identify various types of pairwise compatibility graphs are quite generic and may be useful to discover other properties of these graphs.

Pairwise Compatibility Graphs: A Survey

“…• the union of LPG and mLPG does not coincide with the whole class PCG. Indeed, the class C of cycles is in PCG but does not belong either to LPG or to mLPG [16,55]; • the class T of threshold graphs belongs to LPG ∩ mLPG [20];…”

“…PCGs. Many graph classes have been proved to be in PCG: cycles, single chord cycles, cacti, tree power graphs, Steiner k-power and phylogenetic k-power graphs [54,55]. More recently, even trees, ladder graphs, triangle-free outerplanar 3-graphs [52] and Dilworth 2 graphs [19] have been proved to be PCGs.…”

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.…”

On the Pairwise Compatibility Property of Some Superclasses of Threshold Graphs

“…Due to the apparent difficulty of the pairwise compatibility recognition problem for arbitrary graphs, the research is focussed on the study of this problem for specific classes of graphs. Following this line of research many classes of graphs are proven to be in PCG such as cliques and disjoint union of cliques [1], chordless cycles and single chord cycles [20], ladder graphs [18], some particular subclasses of bipartite graphs [19] and graphs with Dilworth number two [8]. Moreover, it is proven that all graphs with 7 nodes or less are PCGs [17,6], whereas the smallest example of a graph that is not PCG has 8 nodes [10].…”