Locating-dominating sets in twin-free graphs

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $\gamma_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] that if $G$ is a twin-free graph of order $n$ without isolated vertices, then $\gamma_L(G)\le \frac{n}{2}$. We prove the general bound $\gamma_L(G)\le \frac{2n}{3}$, slightly improving over the $\lfloor\frac{2n}{3}\rfloor+1$ bound of Garijo et al. We then provide constructions of graphs reaching the $\frac{n}{2}$ bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees $G$ that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.Comment: 11 pages; 4 figure

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

Finding maximum k-cliques faster using lazy global domination

No abstract available

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs

Maker-Breaker domination number

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $\gamma_{{\rm MB}}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $\gamma_{{\rm MB}}'(G)$. Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant $\gamma_{{\rm MB}}(G)$ is also compared with the domination number. Using the Erd\H{o}s-Selfridge Criterion a large class of graphs $G$ is found for which $\gamma_{{\rm MB}}(G) > \gamma(G)$ holds. Residual graphs are introduced and used to bound/determine $\gamma_{{\rm MB}}(G)$ and $\gamma_{{\rm MB}}'(G)$. Using residual graphs, $\gamma_{{\rm MB}}(T)$ and $\gamma_{{\rm MB}}'(T)$ are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.Comment: 20 pages, 5 figure