Total domination versus paired domination

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by gamma_t . The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Gamma_t . A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by gamma_p . The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Gamma_p . In this paper we prove several results on the ratio of these four parameters: For each r ge 2 we prove the sharp bound gamma_p/gamma_t le 2 - 2/r for K_{1,r} -free graphs. As a consequence, we obtain the sharp bound gamma_p/gamma_t le 2 - 2/(Delta+1) , where Delta is the maximum degree. We also show for each r ge 2 that {C_5,T_r} -free graphs fulfill the sharp bound gamma_p/gamma_t le 2 - 2/r , where T_r is obtained from K_{1,r} by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Gamma_p / Gamma_t . Further, we prove that a graph hereditarily has an induced paired dominating set iff gamma_p le Gamma_t holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio gamma_p / Gamma_t taken over the induced subgraphs of a graph. As a consequence, we prove for each r ge 3 the sharp bound gamma_p/Gamma_t le 2 - 2/r for graphs that do not contain the corona of K_{1,r} as subgraph. In particular, we obtain the sharp bound gamma_p/Gamma_t le 2 - 2/Delta

Total domination versus paired domination

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by gamma_t . The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Gamma_t . A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by gamma_p . The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Gamma_p . In this paper we prove several results on the ratio of these four parameters: For each r ge 2 we prove the sharp bound gamma_p/gamma_t le 2 - 2/r for K_{1,r} -free graphs. As a consequence, we obtain the sharp bound gamma_p/gamma_t le 2 - 2/(Delta+1) , where Delta is the maximum degree. We also show for each r ge 2 that {C_5,T_r} -free graphs fulfill the sharp bound gamma_p/gamma_t le 2 - 2/r , where T_r is obtained from K_{1,r} by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Gamma_p / Gamma_t . Further, we prove that a graph hereditarily has an induced paired dominating set iff gamma_p le Gamma_t holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio gamma_p / Gamma_t taken over the induced subgraphs of a graph. As a consequence, we prove for each r ge 3 the sharp bound gamma_p/Gamma_t le 2 - 2/r for graphs that do not contain the corona of K_{1,r} as subgraph. In particular, we obtain the sharp bound gamma_p/Gamma_t le 2 - 2/Delta

Upper paired domination versus upper domination

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question

Domination parameters with number 2: Interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γw2(G), the 2-domination number, γ2(G), the {2}-domination number, γ{2}(G), the double domination number, γ×2(G), the total {2}-domination number, γt{2}(G), and the total double domination number, γt×2(G), where G is a graph in which the corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G), and two classical parameters, the domination number, γ(G), and the total domination number, γt(G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Brešar, Boštjan. Institute of Mathematics, Physics and Mechanics; Eslovenia. University of Maribor; EsloveniaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Milanič, Martin. University of Primorska; EsloveniaFil: Safe, Martin Dario. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin

Domination parameters with number 2: interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number, $\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number, $\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times 2}(G)$, where $G$ is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, $\gamma_R(G)$, and two classical parameters, the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure

On the existence of total dominating subgraphs with a prescribed additive hereditary property

AbstractRecently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set

Indicated total domination game

A vertex $u$ in a graph $G$ totally dominates a vertex $v$ if $u$ is adjacent to $v$ in $G$. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex of $G$ is totally dominated by a vertex in $S$. The indicated total domination game is played on a graph $G$ by two players, Dominator and Staller, who take turns making a move. In each of his moves, Dominator indicates a vertex $v$ of the graph that has not been totally dominated in the previous moves, and Staller chooses (or selects) any vertex adjacent to $v$ that has not yet been played, and adds it to a set $D$ that is being built during the game. The game ends when every vertex is totally dominated, that is, when $D$ is a total dominating set of $G$. The goal of Dominator is to minimize the size of $D$, while Staller wants just the opposite. Providing that both players are playing optimally with respect to their goals, the size of the resulting set $D$ is the indicated total domination number of $G$, denoted by $\gamma_t^{\rm i}(G)$. In this paper we present several results on indicated total domination game. Among other results we prove the Indicated Total Continuation Principle, and we show that the indicated total domination number of a graph is bounded below by the well studied upper total domination number.Comment: 20 pages, 6 figures, 18 references, corrected a reference and added MSC clas