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Showing papers on "K-tree published in 1997"


Journal ArticleDOI
TL;DR: An efficient algorithm is proposed for solving the clique r-domination problem and the cliques r-packing problem on dually chordal graphs which are a natural generalization of strongly chordalGraphs.
Abstract: Let $\cal C$ be a family of cliques of a graph G=(V,E). Suppose that each clique C of $\cal C$ is associated with an integer r(C)$, where $r(C) \ge 0$. A vertex v r-dominates a clique C of G if $d(v,x) \le r(C)$ for all $x \in C$, where d(v,x) is the standard graph distance. A subset $D \subseteq V$ is a clique r-dominating set of G if for every clique $C \in \cal C$ there is a vertex $u \in D$ which r-dominates C. A clique r-packing set is a subset $P \subseteq \cal C$ such that there are no two distinct cliques $C',C''\in P$ $r$-dominated by a common vertex of G. The clique r-domination problem is to find a clique r-dominating set with minimum size and the clique r-packing problem is to find a clique r-packing set with maximum size. The formulated problems include many domination and clique-transversal-related problems as special cases. In this paper an efficient algorithm is proposed for solving these problems on dually chordal graphs which are a natural generalization of strongly chordal graphs. The efficient algorithm is mainly based on the tree structure and special vertex elimination orderings of dually chordal graphs. In some important particular cases where the algorithm works in linear time the obtained results generalize and improve known results on strongly chordal graphs.

33 citations


Book ChapterDOI
18 Sep 1997
TL;DR: A graph is a cycle of cliques, if its set of vertices can be partitioned into clusters, such that each cluster is a clique and the cliques form a cycle.
Abstract: A graph is a cycle of cliques, if its set of vertices can be partitioned into clusters, such that each cluster is a clique and the cliques form a cycle. Then there is a partition of the set of edges into inner edges of the cliques and interconnection edges between the clusters. Cycles of cliques are a special instance of two-level clustered graphs. Such graphs are drawn by a two phase method: draw the top level graph and then browse into the clusters. In general, it is NP-hard whether or not a graph is a two-level clustered graph of a particular type, e.g. a clique or a planar graph or a triangle of cliques. However, it is efficiently solvable whether or not a graph is a path of cliques or is a large cycle of cliques.

15 citations


Journal ArticleDOI
TL;DR: In this paper, the authors gave polynomial-time algorithms for the majority domination problem in trees, cographs, and $k$-trees with fixed $k.
Abstract: This paper studies algorithmic aspects of majority domination, which is a variation of domination in graph theory. A majority dominating function of a graph $G=(V,E)$ is a function $g$ from $V$ to $\{-1,1\}$ such that $\sum_{u\in N[v]}\,g(u)\geq 1$ for at least half of the vertices $v\in V$. The majority domination problem is to find a majority dominating function $g$ of a given graph $G=(V,E)$ such that $\sum_{v\in V}\,g(v)$ is minimized. The concept of majority domination was introduced by Hedetniemi and studied by Broere {\em et al}., who gave exact values for the majority domination numbers of complete graphs, complete bipartite graphs, paths, and unions of two complete graphs. They also proved that the majority domination problem is NP-complete for general graphs; and asked if the problem NP-complete for trees. The main result of this paper is to give polynomial-time algorithms for the majority domination problem in trees, cographs, and $k$-trees with fixed $k$.

14 citations


Journal ArticleDOI
01 Sep 1997-Networks
TL;DR: In this article, it was shown that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x, y of G d(x, y) ≤ r(x) + r(y) + 1 holds where r : V → N is a given vertex function.
Abstract: In this paper, we consider r-dominating cliques in homogeneously orderable graphs (a common generalization of dually chordal and distance-hereditary graphs) and their relation to strict r-packing sets. We prove that a homogeneously orderable graph G possesses an r-dominating clique if and only if for any pair of vertices x, y of G d(x, y) ≤ r(x) + r(y) + 1 holds where r : V → N is a given vertex function. Furthermore, we show that for homogeneously orderable graphs with r-dominating cliques the cardinality of a maximum strict r-packing set equals the cardinality of a minimum r-dominating clique provided the last parameter is not two. Finally, we present two efficient algorithms: The first one decides whether a given homogeneously orderable graph has an r-dominating clique and, if so, computes both a minimum r-dominating clique and a maximum strict r-packing set of the graph. The second one computes a minimum connected r-dominating set in a homogeneously orderable graph. © 1997 John Wiley & Sons, Inc. Networks 30: 121–131, 1997

5 citations


Journal ArticleDOI
01 Jul 1997-Networks
TL;DR: In this article, the edge coloring of a partial k-tree into two partial p-and q-trees with p, q < k is considered and an algorithm is provided to construct such a coloring with p + q = k.
Abstract: The problem of the edge coloring partial k-tree into two partial p- and q-trees with p, q < k is considered An algorithm is provided to construct such a coloring with p + q = k Usefulness of this result in a Lagrangian decomposition framework to solve certain combinatorial optimization problems is discussed

5 citations


Journal ArticleDOI
TL;DR: The purpose is to elaborate a theory of planar nets or unfoldings for polyhedra, its generalization and extension to polytopes and to combinatorial poly topes, in terms of morphisms of geometries and the adjacency graph of facets.
Abstract: Our purpose is to elaborate a theory of planar nets or unfoldings for polyhedra, its generalization and extension to polytopes and to combinatorial polytopes, in terms of morphisms of geometries and the adjacency graph of facets.

3 citations


Journal ArticleDOI
TL;DR: Two new special families of complete subgraphs of a graph are studied, one of which reduces to the family of minimal vertex separators while the other is empty, which are as fundamental as minimal vertices for chordal graphs.

1 citations


Proceedings ArticleDOI
28 Apr 1997
TL;DR: The class of doubly chordal graphs is a subclass of chordal graph and a superclass of strongly chordal, which arise in many application areas as mentioned in this paper, and it has been shown that the generation of a doubly perfect elimination ordering can be done in O(log/sup 2/n) time using O(nm) processors on the CRCW PRAM model.
Abstract: The class of doubly chordal graphs is a subclass of chordal graphs and a superclass of strongly chordal graphs, which arise in many application areas. Many optimization problems like domination and Steiner tree which are NP-complete on chordal graphs can be solved in polynomial time on doubly chordal graphs. We investigate several characterizations and properties of doubly chordal graphs. Using these properties we show that the recognition of a doubly chordal graph with n vertices and m edges and the generation of a doubly perfect elimination ordering can be done in O(log/sup 2/ n) time using O(nm) processors on the CRCW PRAM model.