Abstract
A graph G is called a pairwise compatibility graph (PCG, for short) if it admits a tuple \((T,w, d_{\min },d_{\max })\) 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 \(d_{\min }\le d_{\max }\) such that G has an edge between two vertices \(u,v\in V\) if and only if the distance between the two leaves u and v in the weighted tree (T, w) is in the interval \([d_{\min }, d_{\max }]\). The tree T is also called a witness tree of the PCG G. The problem of testing if a given graph is a PCG is not known to be NP-hard yet. To obtain a complete characterization of PCGs is a wide open problem in computational biology and graph theory. In the literature, most witness trees admitted by known PCGs are stars and caterpillars. In this paper, we give 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.
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The work is supported by the National Natural Science Foundation of China, under grants 61772115 and 61370071.
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Xiao, M., Nagamochi, H. (2018). Characterizing Star-PCGs. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_42
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