Xue-gang Chen

Upper bounds on the paired-domination number

Abstract A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G , denoted by γ p r ( G ) . In this work, we present several upper bounds on the paired-domination number in terms of the maximum degree, minimum degree, girth and order.

Triangle-free graphs with large independent domination number

Let G be a simple graph of order n and minimum degree @d. The independent domination number i(G) is defined as the minimum cardinality of an independent dominating set of G. We prove the following conjecture due to Haviland [J. Haviland, Independent domination in triangle-free graphs, Discrete Mathematics 308 (2008), 3545-3550]: If G is a triangle-free graph of order n and minimum degree @d, then i(G)@?n-2@d for n/4@?@d@?n/3, while i(G)@?@d for n/3<@d@?2n/5. Moreover, the extremal graphs achieving these upper bounds are also characterized.

A note on independent vertex–edge domination in graphs☆

Abstract The independent vertex–edge domination number and the upper non-enclaving number of a graph G are denoted by i v e ( G ) and Ψ ( G ) , respectively. Boutrig et al. posed the following question: Let G be a connected graph with order n . Is Ψ ( G ) + i v e ( G ) ≤ n ? In this paper, we provide an infinite family of counterexamples. A new relationship between Ψ ( G ) and i v e ( G ) is established. Furthermore, if G is a connected cubic graph, we answer this question in the affirmative.

Constructing connected bicritical graphs with edge-connectivity 2

A graph G is said to be bicritical if the removal of any pair of vertices decreases the domination number of G. For a bicritical graph G with the domination number t, we say that G is t-bicritical. Let @l(G) denote the edge-connectivity of G. In [2], Brigham et al. (2005) posed the following question: If G is a connected bicritical graph, is it true that @l(G)>=3? In this paper, we give a negative answer toward this question; namely, we give a construction of infinitely many connected t-bicritical graphs with edge-connectivity 2 for every integer t>=5. Furthermore, we give some sufficient conditions for a connected 5-bicritical graph to have @l(G)>=3.

A note on α -total domination in cubic graphs

Let G = ( V , E ) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S . For some α with 0 < α ź 1 , a total dominating set S in G is an α -total dominating set if for every vertex v ź V ź S , | N ( v ) ź S | ź α | N ( v ) | . The α -total domination number of G , denoted by γ α t ( G ) , is the minimum cardinality of an α -total dominating set of G . In Henning and Rad (2012), Henning and Rad posed the following question: Let G be a connected cubic graph with order n . Is it true that γ α t ( G ) ź 3 n 4 for 2 3 < α ź 1 ?In this paper, we give a positive answer toward this question. Furthermore, we give a characterization on cubic graphs attaining the bound for the α -total domination number.