A Necessary Condition and a Sufficient Condition for Pairwise Compatibility Graphs

A survey on pairwise compatibility graphs

“…The graph shown in Figure 5(a) has two disjoint chordless cycles and hence by Theorem 11 the complement of the graph is not a PCG. The correctness of Theorem 11 is immediate from the following lemmas [20]. By Theorem 15, the complement of the graph shown in Figure 5(b) is a PCG.…”

“…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.…”

“…Hossain et al gave a necessary condition and a sufficient condition for a graph to be PCG [20]. The necessary condition is stated as follows [20].…”

“…Despite being the first necessary condition and first sufficient condition, there is a gap between the conditions. Hossain et al identified four interesting graph classes between these two conditions as follows [20].…”

Exact-$2$-Relation Graphs

“…Originally introduced in the context of phylogenetics [3], they have received considerable interest in the last years, see [4,5] and the references therein, as well as [6,7,8]. A further generalization of "multi-interval" PCGs in explored in [9].…”