Renu C. Laskar

On domination and independent domination numbers of a graph

Abstract For a graph G , the definitions of domination number, denoted γ( G ), and independent domination number, denoted i ( G ), are given, and the following results are obtained: Theorem. If G does not have an induced subgraph isomorphic to K 1,3 , then γ ( G ) = i ( G ). Corollary 1. For any graph G , γ ( L ( G ))= i ( L ( G )), where L ( G ) is the line graph of G . (This extends the result γ ( L ( T ))= i ( L ( T )), where T is a tree. Hedetniemi and Mitchell, S. E. Conf. Baton Rouge, 1977.) Corollary 2. For any Graph G , γ ( M ( G ))= i ( M ( G )), where M is the middle graph of G .

Bibliography on domination in graphs and some basic definitions of domination parameters

The following bibliography on Domination in Graphs has been compiled over the past six years at Clemson University, where we regularly maintain a computer data base on this topic. Several people have been especially helpful in keeping this bibliography up-to-date and we would like to thank them: E.J. Cockayne, Victoria, British Columbia; P.J. Slater, Huntsville, Alabama; Maciej Syslo, Wroclaw, Poland; Bohdan Zelinka, Liberec, Czechoslovakia; E. Sampathkumar, Dharwad, India; A. Brandstadt, Restock, GDR; and Peter Hammer, New Brunswick, New Jersey. This bibliography essentially starts with the graph theory texts of Kijnig (1950), Berge (1958) and Ore (1962). Although a few research papers on domination were published between 1958 and 1975, a survey paper by Cockayne and Hedetniemi (1975) served to focus attention on the subject sufficiently to ‘get the ball rolling’. By 1988 the domination bibliography included well over 300 citations, about one-third of which are concerned with algorithms for computing various domination numbers for special classes of graphs. In our view, the rapid growth in the number of domination papers is attributable largely to three factors: (i) the diversity of applications to both real-world and other mathematical ‘covering’ or ‘location’ problems; (ii) the wide variety of domination parameters that can be defined; (iii) the NP-completeness of the basic domination problem, its close and ‘natural’ relationships to other NP-complete problems, and the subsequent interest in finding polynomial time solutions to domination problems in special classes of graphs. Thus we expect that this bibliography will continue to grow at a steady rate. As far as we know, only four survey papers have been written on domination in graphs: Cockayne and Hedetniemi, 1975;

On the Algorithmic Complexity of Total Domination

A set of vertices D is a dominating set for a graph

Restrained domination in graphs

G = (V,E)

Irredundancy in circular arc graphs

if every vertex not in D is adjacent to a vertex in D. A set of vertices is a total dominating set if every vertex in V is adjacent to a vertex in D. Cockayne, Goodman and Hedetniemi presented a linear time algorithm to determine minimum dominating sets for trees. Booth and Johnson established the NP-completeness of the problem for undirected path graphs. This paper presents a linear time algorithm to determine minimum total dominating sets of a tree and shows that for undirected path graphs the problem remains NP-complete.

On decomposition of r-partite graphs into edge-disjoint hamilton circuits

Abstract In this paper, we initiate the study of a variation of standard domination, namely restrained domination. Let G =( V , E ) be a graph. A restrained dominating set is a set S ⊆ V where every vertex in V − S is adjacent to a vertex in S as well as another vertex in V − S . The restrained domination number of G , denoted by γ r ( G ), is the smallest cardinality of a restrained dominating set of G . We determine best possible upper and lower bounds for γ r ( G ), characterize those graphs achieving these bounds and find best possible upper and lower bounds for γ r (G)+γ r ( G ) where G is a connected graph. Finally, we give a linear algorithm for determining γ r ( T ) for any tree and show that the decision problem for γ r ( G ) is NP-complete even for bipartite and chordal graphs.

On powers and centers of chordal graphs

Abstract A set of vertices X is called irredundant if for every x in X the closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredundant set equals the size of a maximum independent set. Variants of irredundancy called oo-irredundance, co-irredundance, and oc-irredundancy are defined using combinations of open and closed neighborhoods. We prove that for circular arc graphs the size of a maximum oo-irredundant set equals 2β ∗ or 2β ∗ +1 (depending on parity) where β ∗ is the strong matching number. We also show that for circular arc graphs, the size of a maximum co-irredundant set equals the maximum number of vertices in a set consisting of disjoint K1s and K2s. Similar results are proven for bipartite graphs.

Offensive alliances in graphs

Let K(n;r) denote the complete r-partite graph K(n, n,..., n). It is shown here that for all even n(r - 1) >= 2, K(n;r) is the union of n(r - 1)2 of its Hamilton circuits which are mutually edge-disjoint, and for all odd n(r - 1) >= 1, K(n;r) is the union of (n(r - 1) - 1)2 of its Hamilton circuits and a 1-factor, all of which are mutually edge-disjoint.

A linear algorithm for finding a minimum dominating set in a cactus algorithm for finding a minimum dominating set in a cactus

A graph is chordal if every cycle of length strictly greater than three has a chord. A necessary and sufficient condition is given for all powers of a chordal graph to be chordal. In addition, it is shown that for connected chordal graphs the center (the set of all vertices with minimum eccentricity) always induces a connected subgraph. A relationship between the radius and diameter of chordal graphs is also established.

Linear-time Computability of Combinatorial Problems on Generalized-Series-Parallel Graphs

A set S is an offensive alliance if for every vertex v in its boundary N(S)−S it holds that the majority of vertices in v’s closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, we show that the offensive alliance number is at most 2/3 the order and the strong offensive alliance number is at most 5/6 the order.