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## Interval Graphs (interval + graph)
## Selected Abstracts## Bi-arc graphs and the complexity of list homomorphisms JOURNAL OF GRAPH THEORY, Issue 1 2003Tomas FederAbstract Given graphs G, H, and lists L(v) , V(H), v , V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) , V(H) such that uv , E(G) implies f(u)f(v) , E(H), and f(v) , L(v) for all v , V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) , V(H), v , V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP -complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP -complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi-arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi-arc graph, and is NP -complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61,80, 2003 [source] ## On the L(h, k)-labeling of co-comparability graphs and circular-arc graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2009Tiziana CalamoneriAbstract Given two nonnegative integers h and k, an L(h, k)- labeling of a graph G = (V, E) is a map from V to a set of integer labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h, k)-labeling problem is to produce a legal labeling that minimizes the largest label used. Since the decision version of the L(h, k)-labeling problem is NP-complete, it is important to investigate classes of graphs for which the problem can be solved efficiently. Along this line of thought, in this article we deal with co-comparability graphs, its subclass of interval graphs, and circular-arc graphs. To the best of our knowledge, ours is the first reported result concerning the L(h, k)-labeling of co-comparability and circular-arc graphs. In particular, we provide the first algorithm to L(h, k)-label co-comparability, interval, and circular-arc graphs with a bounded number of colors. Finally, in the special case where k = 1 and G is an interval graph, our algorithm improves on the best previously-known ones using a number of colors that is at most twice the optimum. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009 [source] ## Efficient algorithms for centers and medians in interval and circular-arc graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2002Sergei BespamyatnikhAbstract The p -center problem is to locate p facilities on a network so as to minimize the largest distance from a demand point to its nearest facility. The p -median problem is to locate p facilities on a network so as to minimize the average distance from a demand point to its closest facility. We consider these problems when the network can be modeled by an interval or circular-arc graph whose edges have unit lengths. We provide, given the interval model of an n vertex interval graph, an O(n) time algorithm for the 1-median problem on the interval graph. We also show how to solve the p -median problem, for arbitrary p, on an interval graph in O(pn log n) time and on a circular-arc graph in O(pn2 log n) time. We introduce a spring representation of the objective function and show how to solve the p -center problem on a circular-arc graph in O(pn) time, assuming that the arc endpoints are sorted. © 2002 Wiley Periodicals, Inc. [source] ## Bi-arc graphs and the complexity of list homomorphisms JOURNAL OF GRAPH THEORY, Issue 1 2003Tomas FederAbstract Given graphs G, H, and lists L(v) , V(H), v , V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) , V(H) such that uv , E(G) implies f(u)f(v) , E(H), and f(v) , L(v) for all v , V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) , V(H), v , V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP -complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP -complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi-arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi-arc graph, and is NP -complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61,80, 2003 [source] ## Mixed search number and linear-width of interval and split graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2010Fedor V. FominAbstract We show that the mixed search number and the linear-width of interval graphs and of split graphs can be computed in linear time and in polynomial time, respectively. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## Improper coloring of unit disk graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2009Frédéric HavetAbstract Motivated by a satellite communications problem, we consider a generalized coloring problem on unit disk graphs. A coloring is k -improper if no more than k neighbors of every vertex have the same colour as that assigned to the vertex. The k -improper chromatic number ,k(G) is the least number of colors needed in a k -improper coloring of a graph G. The main subject of this work is analyzing the complexity of computing ,k for the class of unit disk graphs and some related classes, e.g., hexagonal graphs and interval graphs. We show NP-completeness in many restricted cases and also provide both positive and negative approximability results. Because of the challenging nature of this topic, many seemingly simple questions remain: for example, it remains open to determine the complexity of computing ,k for unit interval graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009 [source] ## On the L(h, k)-labeling of co-comparability graphs and circular-arc graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2009Tiziana CalamoneriAbstract Given two nonnegative integers h and k, an L(h, k)- labeling of a graph G = (V, E) is a map from V to a set of integer labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h, k)-labeling problem is to produce a legal labeling that minimizes the largest label used. Since the decision version of the L(h, k)-labeling problem is NP-complete, it is important to investigate classes of graphs for which the problem can be solved efficiently. Along this line of thought, in this article we deal with co-comparability graphs, its subclass of interval graphs, and circular-arc graphs. To the best of our knowledge, ours is the first reported result concerning the L(h, k)-labeling of co-comparability and circular-arc graphs. In particular, we provide the first algorithm to L(h, k)-label co-comparability, interval, and circular-arc graphs with a bounded number of colors. Finally, in the special case where k = 1 and G is an interval graph, our algorithm improves on the best previously-known ones using a number of colors that is at most twice the optimum. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009 [source] ## Approximate L(,1,,2,,,,t)-coloring of trees and interval graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Alan A. BertossiAbstract Given a vector (,1,,2,,,,t) of nonincreasing positive integers, and an undirected graph G = (V,E), an L(,1,,2,,,,t)-coloring of G is a function f from the vertex set V to a set of nonnegative integers such that ,f(u) , f(v), , ,i, if d(u,v) = i, 1 , i , t, where d(u,v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(,1,,2,,,,t)-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(,1,,2,,,,t)-coloring of two relevant classes of graphs,trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t + ,1)) and O(nt2,1) time algorithms are proposed to find ,-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and , is a constant depending on t and ,1,,,,t. Moreover, an O(n) time algorithm is given for the L(,1,,2)-coloring of unit interval graphs, which provides a 3-approximation. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 204,216 2007 [source] |