Dynamic programming on distance-hereditary graphs

Abstract

Abstract. In this paper, we define a one-vertex-extension tree for a distance-hereditary graph and show how to build it. We then give a unified approach to designing efficient dynamic programming algorithms for distance-hereditary graphs based upon the one-vertex-extension tree. We give linear time algorithms for the weighted vertex cover and weighted independent domination problems and give an O(n 2) time algorithm to compute a minimum fill-in and the treewidth for a distance-hereditary graph. IntroductionThe distance da (u, v) between two vertices u and v of a connected graph G is the minimum length of a u-v path in G. A graph is distance-hereditary if each pair of vertices are equidistant in every connected induced subgraph containing them. Distance-hereditary graphs form a subclass of perfect graphs [7,11,12] that are graphs G in which the maximum clique size equals the chromatic number for every induced subgraph of G. Properties and optimization problems in distance-hereditary graphs have been extensively studied during the past two decades [2,3, 4,7,8,9,11,12,17,19,20] which result in sequential algorithms that solve quite a few graph-theoretical problems on this special class of graphs. However, some problems remain unresolved on distance-hereditary graphs. For example, the independent domination problem and the vertex cover problem, the minimum fill-in and treewidth problem, and the edge domination problem etc. Most of the known polynomial time algorithms for distance-hereditary graphs utilize ad hoc techniques. In this paper, we first define a new data structure, called one-vertex-exlension tree, to represent a distance-hereditary graph in the form of a rooted tree. We give a new recursive definition for this class of graphs. We then show how to design dynamic programming algorithms based on this recursive definition. We present linear time algorithms to solve the weighted vertex cover problem and weighted independent domination problem and present a O(n 2) time algorithm to compute the minimum fill-in and treewidth for a distance-hereditary graph. The above three problems are NP-complete for general graphs [1,5,14], and the vertex cover problem is known to be MAX SNP-

Exponential Bounds on Graph Enumerations from Vertex Incremental Characterizations

“…• Pick a target vertex x in G, add a new vertex x to G: 5 We may be applying operations in a slightly different order than given in constructing the tree. As long as we read the tree in level order, this does not change the underlying graph.…”

“…We call these trees vertex incremental trees [5], and they are structures that encode the vertex incremental operations used to construct the corresponding graphs. Historically, this idea first emerged in the enumeration of cographs [11].…”

Weighted Maximum-Clique Transversal Sets of Graphs

“…Chang et al 44 showed that distancehereditary graphs can be defined, recursively. By Theorem 8.1, a distance-hereditary graph G has its own twin set TS G , the twin set TS G is a subset of vertices of G, and it is defined recursively.…”

Laminar Structure of Ptolemaic Graphs and Its Applications

“…The other nodes have margin 0, and the other arcs have distribution 0. Hence the graph in Figure 5 has a Hamiltonian cycle, e.g., (1,8,2,9,3,10,4,11,5,14,16,15,7,12,6,13,1).…”

“…Especially, Bandelt and Mulder showed that any distance hereditary graph can be obtained from K 2 by a sequence of extensions called "adding a pendant vertex" and "splitting a vertex." Using the characterizations, many efficient algorithms have been found for distance hereditary graphs [6,2,5,21,17,7]. However, the recognition of distance hereditary graphs in linear time is not so simple; Hammer and Maffray's algorithm [14] fails in some cases, and Damiand, Habib, and Paul's algorithm [9] requires to build a cotree in linear time (see [9,Chapter 4] for further details), where the cotree can be constructed in linear time by using recent algorithm based on multisweep LBFS approach by Bretscher, Corneil, Habib, and Paul [4].…”