Upper bounds for domination related parameters in graphs on surfaces

AbstractIn 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

Total Roman {2}-domination in graphs

[EN] Given a graph G = (V, E), a function f: V -> {0, 1, 2} is a total Roman {2}-dominating function if every vertex v is an element of V for which f (v) = 0 satisfies that n-ary sumation (u)(is an element of N (v)) f (v) >= 2, where N (v) represents the open neighborhood of v, and every vertex x is an element of V for which f (x) >= 1 is adjacent to at least one vertex y is an element of V such that f (y) >= 1. The weight of the function f is defined as omega(f ) = n-ary sumation (v)(is an element of V) f (v). The total Roman {2}-domination number, denoted by gamma(t)({R2})(G), is the minimum weight among all total Roman {2}-dominating functions on G. In this article we introduce the concepts above and begin the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other domination related parameters in graphs. In addition, we prove that the complexity of computing the value gamma(t)({R2})(G) is NP-hard, even when restricted to bipartite or chordal graphsCabrera García, S.; Cabrera Martinez, A.; Hernandez Mira, FA.; Yero, IG. (2021). Total Roman {2}-domination in graphs. Quaestiones Mathematicae. 44(3):411-444. https://doi.org/10.2989/16073606.2019.1695230S41144444

NEW DOCTORAL DEGREES Domination numbers of simple polygonal chains and multiple linear hexagonal chains

Domination is an area in graph theory with an extensive research activity, together with its numerous generalizations and modifications, motivated by various applications and problems. The most interesting part of this area is the multitude of types of domination. In general, a dominating set of a graph is a set $D$ such that every vertex of a graph is either in $D$ or is adjacent to some vertex in $D$. Domination number is the cardinality of the smallest dominating set $D$. The problem to determine the domination number of graph is $NP-hard$ even when restricted to some simple graph structures. However, there are certain graph structures for which domination numbers can be determined by using some well-known mathematical tools like mathematical induction or partition of graph into small parts, mutually isomorphic subgraphs, for which domination number can be easily established. This thesis is focused on distance $k$-domination and total domination on two types of graphs: simple polygonal chains (with cactus chains included) and multiple linear hexagonal chains. After determination of domination numbers of equidistant cactus chains, extremal chains regarding this graph invariant are found. Some results about edge domination are presented for hexagonal cactus chains and then compared to (vertex) domination. Some results were presented about domination ratio. For simple polygonal chains considered types of domination are also investigated. At last, the domination numbers of multiple linear hexagonal chains were determined. The most important part of the thesis is proof that considered dominating set has minimum cardinality among all dominating sets of considered graph. Usual procedure is to either establish that dominating set is perfect, or try to find some obvious property that every (and therefore the minimum) dominating set satisfies or do the partition of graph into smaller parts, called blocks, on which the domination number can be easily established

The Algorithmic Complexity of Bondage and Reinforcement Problems in bipartite graphs

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. The domination number of $G$, denoted by $\gamma(G)$, is the smallest cardinality of a dominating set of $G$. The bondage number of a nonempty graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with domination number larger than $\gamma(G)$. The reinforcement number of $G$ is the smallest number of edges whose addition to $G$ results in a graph with smaller domination number than $\gamma(G)$. In 2012, Hu and Xu proved that the decision problems for the bondage, the total bondage, the reinforcement and the total reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results to bipartite graphs.Comment: 13 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1109.1657; and text overlap with arXiv:1204.4010 by other author

Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016