Total [1,2]-domination in graphs

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by $\gamma_{t[1,2]}(G)$. We establish two sharp upper bounds on the total [1,2]-domination number of a graph $G$ in terms of its order and minimum degree, and characterize the corresponding extremal graphs achieving these bounds. Moreover, we give some sufficient conditions for a graph without total $[1,2]$-set and for a graph with the same total $[1,2]$-domination number, $[1,2]$-domination number and domination number.Comment: 17 page

Hypo-efficient domination and hypo-unique domination

For a graph $G$ let $\gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$\mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets,but $G-v$ has a unique minimum dominating set for each $v\in V(G)$. We show that each hypo-$\mathcal{UD}$ graph $G$ of order at least $3$ is connected and $\gamma(G-v) < \gamma(G)$ for all $v \in V(G)$. We obtain a tight upper bound on the order of a hypo-$\mathcal{P}$ graph in terms of the domination number and maximum degree of the graph, where $\mathcal{P} \in \{\mathcal{UD}, \mathcal{ED}\}$. Families of circulant graphs which achieve these bounds are presented. We also prove that the bondage number of any hypo-$\mathcal{UD}$ graph is not more than the minimum degree plus one.Comment: 13 pages, 1 figur

Bounding and approximating minimum maximal matchings in regular graphs

The edge domination number $\gamma_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on $\gamma_e(G)$ for regular and non-regular graphs $G$ in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs

The bondage number of graphs on topological surfaces: degree-S vertices and the average degree

The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. An orientable surface $\mathbb{S}_h$ of genus $h$, $h \geq 0$, is obtained from the sphere $\mathbb{S}_0$ by adding $h$ handles. A non-orientable surface $\mathbb{N}_q$ of genus $q$, $q \geq 1$, is obtained from the sphere by adding $q$ crosscaps. The Euler characteristic of a surface is defined by $\chi(\mathbb{S}_h) = 2 - 2h$ and $\chi(\mathbb{S}_q)= 2-q$. Let $G$ be a connected graph of order $n$ which is 2-cell embedded on a surface $\mathbb{M}$ with $\chi(\mathbb{M})= \chi$. We prove that $b(G) \leq 7+i$ when $\mathbb{M} = \mathbb{N}_i$, $i=1,2,3$, and $b(G) \leq 12$ when $\mathbb{M} \in \{\mathbb{N}_4, \mathbb{S}_2\}$. We give new arguments that improve the known upper bounds on the bondage number at least when $-7\chi/(\delta(G) - 5) < n \leq -12\chi$, $\delta(G) \geq 6$, where $\delta(G)$ is the minimum degree of $G$. We obtain sufficient conditions for the validity of the inequality $b(G) \leq 2s-2$, provided $G$ has degree $s$ vertices. In particular, we prove that if $\delta (G) = \delta \geq 6$, $\chi \leq -1$ and $-14\chi < \delta - 4 + 2(\delta -5)n$ then $b(G) \leq 2\delta -2$. We show that if $\gamma (G) = \gamma \not = 2$, where $\gamma (G)$ is the domination number of $G$, then $n \geq \gamma + (1 + \sqrt{9+8\gamma-8\chi})/2$; the bound is tight. We also present upper bounds for the bondage number of graphs in terms of the girth, domination number and Euler characteristic. As a corollary we prove that if $\gamma(G) \geq 4$ and $\chi \leq -1$, then $b(G) \leq 11 - 24\chi/(9 + \sqrt{41 - 8\chi})$. Several unanswered questions are posed.Comment: 17 pages, 3 figure

On Bondage Numbers of Graphs -- a survey with some comments

The bondage number of a nonempty graph $G$ is the cardinality of a smallest edge set whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.Comment: 80 page; 14 figures; 120 reference

Lower Bounds for the Domination Numbers of Connected Graphs without Short Cycles

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least $\frac{1}{2}(3+\sqrt{8(m-n)+9})$, where $n$ is the number of vertices in the graph and $m$ is the number of edges in the graph. For graphs with minimum degree $2$ and girth at least $7$, the lower bound can be improved to $\max{\{\sqrt{n}, \sqrt{\frac{2m}{3}}\}}$, where $n$ and $m$ are the numbers of vertices and edges in the graph respectively. In cases where the graph is of minimum degree $2$ and its girth $g$ is at least $12$, the lower bound can be further improved to $\max{\{\sqrt{n}, \sqrt{\frac{\lfloor \frac{g}{3} \rfloor-1}{3}m}\}}$

Bounds on the connected forcing number of a graph

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph

Upper bounds for domination related parameters in graphs on surfaces

In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree.Comment: 10 page

Directed Domination in Oriented Graphs

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $\gamma(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $\Gamma_d(G)$, which is the maximum directed domination number $\gamma(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. We extend this notion to directed domination of all graphs. If $\alpha$ denotes the independence number of a graph $G$, we show that if $G$ is a bipartite graph, we show that $\Gamma_d(G) = \alpha$. We present several lower and upper bounds on the directed domination number.Comment: 18 page

Partial domination - the isolation number of a graph

We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed neighborhood of $D$. Furthermore, the bound is sharp. This seems to be the first result in the direction of partial domination with constrained structure on the graph induced by the non-dominated vertices, which we further elaborate in this paper.Comment: 28 page