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Graph classes related to chordal graphs and chordal bipartite graphs
Electronic Notes in Discrete Mathematics 27 (2006) 73–74 www.elsevier.com/locate/endm
Graph classes related to chordal graphs and chordal bipartite graphs Ngoc Tuy Nguyen 1 Hong Duc University Thanh Hoa, Vietnam
J¨org Bornemann University of Rostock Rostock, Germany
Van Bang Le University of Rostock Rostock, Germany
Keywords: graph classes, induced cycles, tree spanners
Many well-known graph classes are defined by allowing only induced cycles of certain lengths. For instance, forests are exactly those graphs in which no (induced) cycles exist, chordal graphs are exactly those graphs in which every induced cycle has length three, and chordal bipartite graphs are exactly those graphs in which every induced cycle has length four. Let k ≥ 1 be an integer. A graph is called an ICk-graph if every Induced Cycle of it has length k. The class of all ICk-graphs is denoted by ICk. Thus, 1
Email: [email protected]
1571-0653/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.endm.2006.08.062
74
N.T. Nguyen et al. / Electronic Notes in Discrete Mathematics 27 (2006) 73–74
IC1 = IC2 is the class of all forests (we consider graphs without loops and multiple edges), IC3 is the class of all chordal graphs, and IC4 coincides with the class of all chordal bipartite graphs. For k ≥ 5, we give a characterization of ICk-graphs which leads to a simple linear time recognition for ICk-graphs. It also follows from the characterization that ICk-graphs have tree-width at most two, and therefore are series-parallel graphs. We then give two characterizations of all ICk-graphs that admit a tree t-spanner; one of them in terms of forbidden induced subgraphs. (For a given integer t ≥ 1, a tree t-spanner of a graph G is a spanning tree T of G such that the distance between every two vertices in T is at most t times their distance in G.) As a result, there is a simple way to decide in linear time whether or not an ICk-graph admits a tree t-spanner.
Graph classes related to chordal graphs and chordal bipartite graphs Ngoc Tuy Nguyen 1 Hong Duc University Thanh Hoa, Vietnam
J¨org Bornemann University of Rostock Rostock, Germany
Van Bang Le University of Rostock Rostock, Germany
Keywords: graph classes, induced cycles, tree spanners
Many well-known graph classes are defined by allowing only induced cycles of certain lengths. For instance, forests are exactly those graphs in which no (induced) cycles exist, chordal graphs are exactly those graphs in which every induced cycle has length three, and chordal bipartite graphs are exactly those graphs in which every induced cycle has length four. Let k ≥ 1 be an integer. A graph is called an ICk-graph if every Induced Cycle of it has length k. The class of all ICk-graphs is denoted by ICk. Thus, 1
Email: [email protected]
1571-0653/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.endm.2006.08.062
74
N.T. Nguyen et al. / Electronic Notes in Discrete Mathematics 27 (2006) 73–74
IC1 = IC2 is the class of all forests (we consider graphs without loops and multiple edges), IC3 is the class of all chordal graphs, and IC4 coincides with the class of all chordal bipartite graphs. For k ≥ 5, we give a characterization of ICk-graphs which leads to a simple linear time recognition for ICk-graphs. It also follows from the characterization that ICk-graphs have tree-width at most two, and therefore are series-parallel graphs. We then give two characterizations of all ICk-graphs that admit a tree t-spanner; one of them in terms of forbidden induced subgraphs. (For a given integer t ≥ 1, a tree t-spanner of a graph G is a spanning tree T of G such that the distance between every two vertices in T is at most t times their distance in G.) As a result, there is a simple way to decide in linear time whether or not an ICk-graph admits a tree t-spanner.