456 research outputs found

    A Characterization of Signed Graphs with Generalized Perfect Elimination Orderings

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    An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those graphs.Comment: 18 pages; (v2) Reference updated (v3) Major update including title change, shortening of proof of main theorem, addition of applications of main theorem to special cases, reference updat

    Efficient generation of elimination trees and graph associahedra

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    An elimination tree for a connected graph~GG is a rooted tree on the vertices of~GG obtained by choosing a root~xx and recursing on the connected components of~G−xG-x to produce the subtrees of~xx. Elimination trees appear in many guises in computer science and discrete mathematics, and they encode many interesting combinatorial objects, such as bitstrings, permutations and binary trees. We apply the recent Hartung-Hoang-M\"utze-Williams combinatorial generation framework to elimination trees, and prove that all elimination trees for a chordal graph~GG can be generated by tree rotations using a simple greedy algorithm. This yields a short proof for the existence of Hamilton paths on graph associahedra of chordal graphs. Graph associahedra are a general class of high-dimensional polytopes introduced by Carr, Devadoss, and Postnikov, whose vertices correspond to elimination trees and whose edges correspond to tree rotations. As special cases of our results, we recover several classical Gray codes for bitstrings, permutations and binary trees, and we obtain a new Gray code for partial permutations. Our algorithm for generating all elimination trees for a chordal graph~GG can be implemented in time~\cO(m+n) per generated elimination tree, where mm and~nn are the number of edges and vertices of~GG, respectively. If GG is a tree, we improve this to a loopless algorithm running in time~\cO(1) per generated elimination tree. We also prove that our algorithm produces a Hamilton cycle on the graph associahedron of~GG, rather than just Hamilton path, if the graph~GG is chordal and 2-connected. Moreover, our algorithm characterizes chordality, i.e., it computes a Hamilton path on the graph associahedron of~GG if and only if GG is chordal

    Separability and Vertex Ordering of Graphs

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    Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

    The world of hereditary graph classes viewed through Truemper configurations

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    In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

    Party Formation and Policy Outcomes Under Different Electoral Systems

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    This paper provides a game-theoretic model of representative democracy with endogenous party formation. Coalition formation may occur before and after elections, and the expected payoffs from the after-election majority game affect incentives-to form parties before the elections. In this way Duverger\u27s hypothesis can be formally explained by the strategic behavior of political elites. If politicians care primarily about private benefits, the equilibrium policy outcome under a proportional electoralsystem coincides with the median party\u27s position. On the other hand, with quasilinear utility, the.distance from the median voter outcome may be lower with plurality rule
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