Discovering Pairwise Compatibility Graphs

Abstract

Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.

Characterizing Star-PCGs

“…A natural question was whether all graphs are PCGs. This was proposed as a conjecture in [12], and was confuted in [18] by giving a counterexample of a bipartite graph with with 15 vertices. Later, a counterexample with eight vertices and a counterexample of a planar graph with 20 vertices were found [9].…”

“…Few methods have been known for constructing a corresponding tuple (T, w, d min , d max ) from a given graph G. The inverse problem attracts certain interests in graph algorithms, which may also have potential applications in computational biology. It has been extensively studied from many aspects after the introduction of PCG [3,6,7,9,19,18].…”

“…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].…”

Some reduction operations to pairwise compatibility graphs

“…Not every graph is a PCG. Yanhaona, Bayzid and Rahman [13] constructed the first non-PCG, which is a bipartite graph with 15 vertices. Later, an example with 8 vertices was found in [8].…”

“…Every cycle with at most one chord has also been shown to be a PCG [14]. Other subclasses of graphs currently known as PCGs are power graphs of trees [13], threshold graphs [5], triangle-free outerplanar 3-graphs [12], a special subclasses of split matrogenic graphs [6], Dilworth 2 graphs [3,4], the complement of a forest [9], the complement of a cycle [2] and so on. Some conditions for a graph not being a PCG have also been developed [8,9,13,10].…”