Independent domination in chordal graphs

A graph chordal if it does not contain any cycle of length greater than three as an induced subgraph. A set of S of vertices of a graph G = (V,E) is independent if not two vertices in S are adjacent, and is dominating if every vertex in V?S is adjacent to some vertex in S. We present a linear algorithm to locate a minimum weight independent dominating set in a chordal graph with 0–1 vertex weights.     

Independent domination in triangle-free graphs

Let G be a simple graph of order n and minimum degree δ. The independent domination numberi(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds, as functions of n and δ?n/2, for the independent domination number of triangle-free graphs, and over part of the range achieve best possible results.     

Independent perfect domination sets in Cayley graphs

In this paper, we show that a Cayley graph for an abelian group has an independent perfect domination set if and only if it is a covering graph of a complete graph. As an application, we show that the hypercube Q_n has an independent perfect domination set if and only if Q_n is a regular covering of the complete graph K_n+1 if and only if n = 2^m ? 1 for some natural number m. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 213–219, 2001     

Independent domination in subcubic graphs of girth at least six

A set $S$ of vertices in a graph $G$ is an independent dominating set of $G$ if $S$ is an independent set and every vertex not in $S$ is adjacent to a vertex in $S$. The independent domination number, $i\left(G\right)$, of $G$ is the minimum cardinality of an independent dominating set. In this paper, we extend the work of Henning, Löwenstein, and Rautenbach (2014) who proved that if $G$ is a bipartite, cubic graph of order $n$ and of girth at least $6$, then $i\left(G\right)\le \frac{4}{11}n$. We show that the bipartite condition can be relaxed, and prove that if $G$ is a cubic graph of order $n$ and of girth at least $6$, then $i\left(G\right)\le \frac{4}{11}n$.     

Balloons,cut‐edges,matchings, and total domination in regular graphs of odd degree

A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}$ be the family of connected (2r+1)‐regular graphs with n vertices, and let ${{b}}={{max}}\{{{b}}({{G}}): {{G}}\in {\mathcal{F}}_{{{n}},{{r}}}\}$. For ${{G}}\in{\mathcal{F}}_{{{n}},{{r}}}$, we prove the sharp inequalities c(G)?[r(n?2)?2]/(2r²+2r?1)?1 and α′(G)?n/2?rb/(2r+1). Using b?[(2r?1)n+2]/(4r²+4r?2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ_t(G) of a cubic graph, we prove γ_t(G)?n/2?b(G)/2 (except that γ_t(G) may be n/2?1 when b(G)=3 and the balloons cover all but one vertex). With α′(G)?n/2?b(G)/3 for cubic graphs, this improves the known inequality γ_t(G)?α′(G). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 116–131, 2010     

Power domination in graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number γ_P(G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.     

Set domination in graphs

Let G = (V, E) be a connected graph. A set D ? V is a set-dominating set (sd-set) if for every set T ? V ? D, there exists a nonempty set S ? D such that the subgraph 〈S ∪ T〉 induced by S ∪ T is connected. The set-domination number γ_s(G) of G is the minimum cardinality of a sd-set. In this paper we develop properties of this new parameter and relate it to some other known domination parameters.     

Line domination in graphs

A setS of lines is a line dominating set if every line not inS is adjacent to some line ofS. The line domination number of a graph is the cardinality of a minimum line dominating set. In this paper we study the line dominating sets and obtain bounds for the line domination number. Also, Nordhaus-Gaddum type results are obtained for the line domination number and the line domatic number.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for $k$-regular graphs with $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.     

Vertex-edge domination in graphs

In this paper we study graph parameters related to vertex-edge domination, where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. First, we present new relationships relating the ve-domination to some other domination parameters, answering in the affirmative four open questions posed in the 2007 PhD thesis by Lewis. Then we provide an upper bound for the independent ve-domination number in terms of the ve-domination number for every nontrivial connected K_1,k-free graph, with k ≥ 3, and we show that the independent ve-domination number is bounded above by the domination number for every nontrivial tree. Finally, we establish an upper bound on the ve-domination number for connected C₅-free graphs, improving a recent bound given for trees.     

Vertex-edge domination in graphs

Aequationes mathematicae - We establish that for any connected graph G of order $$n \ge 6$$ , a minimum vertex-edge dominating set of G has at most n/3 vertices, thus affirmatively answering the...     

Maximum induced matchings in graphs

We provide a formula for the number of edges of a maximum induced matching in a graph. As applications, we give some structural properties of (k + 1)K₂-free graphs, construct all 2K₂-free graphs, and count the number of labeled 2K₂-free connected bipartite graphs.