Characterizing Star-PCGs

A graph G is called a pairwise compatibility graph (PCG, for short) if it admits a tuple $$(T,w, d_{\min },d_{\max })$$ ( T , w , d min , d max ) of a tree T whose leaf set is equal to the vertex set of G , a non-negative edge weight w , and two non-negative reals $$d_{\min }\le d_{\max }$$ d min ≤ d max such that G has an edge between two vertices $$u,v\in V$$ u , v ∈ V if and only if the distance between the two leaves u and v in the weighted tree ( T ,  w ) is in the interval $$[d_{\min }, d_{\max }]$$ [ d min , d max ] . The tree T is also called a witness tree of the PCG G . How to recognize PCGs is a wide-open problem in the literature. This paper gives a complete characterization for a graph to be a star-PCG (a PCG that admits a star as its witness tree), which provides us the first polynomial-time algorithm for recognizing star-PCGs.