Distance-hereditary graphs, Steiner trees, and connected domination
01 Jun 1988-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 17, Iss: 3, pp 521-538
TL;DR: It is shown that the problems of finding cardinality Steiner trees and connected dominating sets are polynomially solvable in a distance-hereditary graph.
Abstract: Distance-hereditary graphs have been introduced by Howorka and studied in the literature with respect to their metric properties. In this paper several equivalent characterizations of these graphs are given: in terms of existence of particular kinds of vertices (isolated, leaves, twins) and in terms of properties of connections, separators, and hangings. Distance-hereditary graphs are then studied from the algorithmic viewpoint: simple recognition algorithms are given and it is shown that the problems of finding cardinality Steiner trees and connected dominating sets are polynomially solvable in a distance-hereditary graph.
Citations
More filters
TL;DR: A survey up to 1989 on the Steiner tree problems which include the four important cases of euclidean, rectilinear, graphic, phylogenetic and some of their generalizations.
Abstract: We give a survey up to 1989 on the Steiner tree problems which include the four important cases of euclidean, rectilinear, graphic, phylogenetic and some of their generalizations. We also provide a rather comprehensive and up-to-date bibliography which covers more than three hundred items.
573 citations
TL;DR: The clique–width of perfect graph classes is studied to see the border within the hierarchy of perfect graphs between classes whose clique-width is bounded and classes whoseClique– width is unbounded.
Abstract: Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every $n\in {\mathcal N}$ there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, $n\in {\mathcal N}$, n ≥ 3, has clique–width exactly n+1.
257 citations
TL;DR: The following bibliography on Domination in Graphs has been compiled over the past six years at Clemson University, where it is expected that this bibliography will continue to grow at a steady rate.
157 citations
01 Jan 1998
TL;DR: This chapter intents to survey the domination problem in graph theory, which is a natural model for many location problems in operations research, from an algorithmic point of view.
Abstract: Graph theory was founded by Euler [78] in 1736 as a generalization to the solution of the famous problem of the Konisberg bridges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real world problems, see the historic book by Biggs, Lloyd and Wilson [19]. This chapter intents to survey the domination problem in graph theory, which is a natural model for many location problems in operations research, from an algorithmic point of view.
105 citations
TL;DR: It is proved that the connected domination number of G, denoted $\gamma_c(G)', is the minimum cardinality of a connected dominating set and two algorithms that construct a set this good are presented.
Abstract: Let G=(V,E) be a connected graph. A connected dominating set $S \subset V$ is a dominating set that induces a connected subgraph of G. The connected domination number of G, denoted $\gamma_c(G)$, is the minimum cardinality of a connected dominating set. Alternatively, $|V|-\gamma_c(G)$ is the maximum number of leaves in a spanning tree of $G$. Let $\delta$ denote the minimum degree of G. We prove that $\gamma_c(G) \leq |V| \frac{\ln(\delta+1)}{\delta+1}(1+o_\delta(1))$. Two algorithms that construct a set this good are presented. One is a sequential polynomial time algorithm, while the other is a randomized parallel algorithm in RNC.
104 citations
Cites background from "Distance-hereditary graphs, Steiner..."
...Results on the connected domination number appear in [3, 4, 7, 8, 10, 11, 16, 17, 23, 25, 26, 28]....
[...]
References
More filters
TL;DR: Distance-hereditary graphs (sensu Howorka) as mentioned in this paper are connected graphs in which all induced paths are isometric, and can be characterized in terms of the distance function d or via forbidden isometric subgraphs.
458 citations
TL;DR: This paper investigates the complexity status of these problems on various sub-classes of perfect graphs, including comparability graphs, chordal graphs, bipartite graphs, split graphs, cographs and κ-trees, where the k-cluster problem is polynomial and the weighted and connected versions are studied.
234 citations