Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.
We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result a...We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].展开更多
A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set ...A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].展开更多
The edge-domsaturation number ds'(G) of a graph G = (V, E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. In this paper, we characterize unicyclic graphs...The edge-domsaturation number ds'(G) of a graph G = (V, E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. In this paper, we characterize unicyclic graphs G with ds'(G) = q – Δ'(G) + 1 and investigate well-edge dominated graphs. We further define γ'–-critical, γ'+-critical, ds'–-critical, ds'+-critical edges and study some of their properties.展开更多
A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a ...A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.展开更多
文摘Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.
文摘We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].
文摘A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].
文摘The edge-domsaturation number ds'(G) of a graph G = (V, E) is the least positive integer k such that every edge of G lies in an edge dominating set of cardinality k. In this paper, we characterize unicyclic graphs G with ds'(G) = q – Δ'(G) + 1 and investigate well-edge dominated graphs. We further define γ'–-critical, γ'+-critical, ds'–-critical, ds'+-critical edges and study some of their properties.
基金supported by Korea Research Foundation Grant (KRF-2002-015-cp0050)the National Natural Science Foundation of China (Grant Nos. 60773078, 10571117)+3 种基金the ShuGuang Plan of Shanghai Education Development Foundation (Grant No. 06SG42)M. A. Henning is supported in part by the South African National Research Foundationthe University of KwaZulu-Natalsupported by Shanghai Leading Academic Discipline Project (No. $30104)
文摘A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.
