Tree Powers

doi:10.1006/jagm.1998.9999

Journal of Algorithms

Volume 29, Issue 1, October 1998, Pages 111-131

https://doi.org/10.1006/jagm.1998.9999 Get rights and content

Abstract

We present the first polynomial algorithm for recognizing tree powers. A graphGis atree powerif there is a treeTand a positive integerksuch thatT^k ≅ G, wherexandyare adjacent inT^kif and only ifd_T(x, y) ≤ k. We also show that a natural extension of tree power recognition is NP-complete, namely, given a graphGand a positive integerr, determine if there is a tree power withinredges ofG.

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Citation Excerpt :
Nevertheless, researchers interested in computational complexity theory are fascinated by these graphs because the clique problem is known to be polynomially solvable for PCGs once the pairwise compatibility tree is provided [16] (while it is well known that, for a general graph, finding whether there is a clique of a given size is NP-complete [13]). Moreover, the class of PCGs appears interesting by itself from a graph theoretic point of view; nowadays, there are a few results proving that some special classes of graphs are PCGs [1,15,17,19,20] but only two results for general graphs, one affirming that all graphs with a number of nodes not greater than 7 are PCGs [6,18] and the other one proving that not all graphs are PCGs, by means of two graphs that cannot be a PCG, one with 15 nodes [20] and the other one with 8 nodes [11]. The last section proposes some conclusions and open problems arisen from this work.
A graph $G = (V, E)$ is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge-weight function w on T, and two non-negative real numbers $d_{m i n}$ and $d_{m a x}$ , $d_{m i n} ⩽ d_{m a x}$ , such that V coincides with the set of leaves of T, and there is an edge $(u, v) \in E$ if and only if $d_{m i n} ⩽ d_{T, w} (u, v) ⩽ d_{m a x}$ where $d_{T, w} (u, v)$ is the sum of the weights of the edges on the unique path from u to v in T. When the constraints on the distance between the pairs of leaves concern only $d_{m a x}$ or only $d_{m i n}$ the two subclasses LPGs (Leaf Power Graphs) and mLPGs (minimum Leaf Power Graphs) are defined.
The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another.
It is known that $LPG \cap mLPG$ is not empty and that threshold graphs, i.e. Dilworth one graphs, are contained in it. In this paper we prove that Dilworth two graphs belong to the set $LPG \cap mLPG$ , too. Our proof is constructive since we show how to compute all the parameters T, w, $d_{m a x}$ and $d_{m i n}$ exploiting the usual representation of Dilworth two graphs in terms of node weight function and thresholds. For graphs with Dilworth number two that are also split graphs, i.e. split permutation graphs, we provide another way to compute T, w, $d_{m i n}$ and $d_{m a x}$ when these graphs are given in terms of their intersection model.
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The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another one. In this paper we give a new characterization of Dilworth k graphs, for each value of k, exactly defining their structure. Moreover, we put these graphs in relation with pairwise compatibility graphs (PCGs), i.e. graphs on n nodes that can be generated from an edge-weighted tree T that has n leaves, each representing a different node of the graph; two nodes are adjacent in the graph if and only if the weighted distance between the corresponding leaves in T is contained between two given non-negative real numbers, m and M. When either m or M are not used to eliminate edges from G, the two subclasses leaf power and minimum leaf power graphs (LPGs and mLPGs, respectively) are defined. Here we prove that graphs that are either LPGs or mLPGs of trees obtained connecting the centers of k stars with a path are Dilworth k graphs, and the opposite is true when $k = 1, 2$ , but not when $k \geq 3$ . Finally, we show that the relations we proved between Dilworth k graphs and chains of k stars hold only for LPGs and mLPGs, but not for PCGs.
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A graph $G = (V, E)$ is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge-weight function w on T, and two non-negative real numbers $d_{m i n}$ and $d_{m a x}$ , $d_{m i n} ⩽ d_{m a x}$ , such that V coincides with the set of leaves of T, and there is an edge $(u, v) \in E$ if and only if $d_{m i n} ⩽ d_{T, w} (u, v) ⩽ d_{m a x}$ where $d_{T, w} (u, v)$ is the sum of the weights of the edges on the unique path from u to v in T. When the constraints on the distance between the pairs of leaves concern only $d_{m a x}$ or only $d_{m i n}$ the two subclasses LPGs (Leaf Power Graphs) and mLPGs (minimum Leaf Power Graphs) are defined.
It is known that $LPG \cap mLPG$ is not empty and that threshold graphs are one of the classes contained in it. It is also known that threshold graphs are all the graphs with Dilworth number one, where the Dilworth number of a graph is the size of the largest subset of its vertices in which the close neighborhood of no vertex contains the neighborhood of another.
In this paper we do one step ahead toward the comprehension of the structure of the set $LPG \cap mLPG$ , proving that graphs with Dilworth number two belong to this intersection.
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