Alliances and related parameters in graphs

In this paper, we show that several graph parameters are known in different areas under completely different names. More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence, positive influence domination, and a parameter associated to fast information propagation in networks to parameters related to various notions of global $r$-alliances in graphs. We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterize various known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination. Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

On The Signed Edge Domination Number of Graphs

Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $n (n \geq 2),$ $\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer $m$, there exists an $m$-connected graph $G$ such that $\gamma'_s(G)\leq -\frac{m}{6}|V(G)|.$ Also for every two natural numbers $m$ and $n$, we determine $\gamma'_s(K_{m,n})$, where $K_{m,n}$ is the complete bipartite graph with part sizes $m$ and $n$

On the Signed (Total) kk-Domination Number of a Graph

Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\rightarrow \{-1,1\}$ is called a \emph{signed $k$-dominating function} of $G$ if $\sum_{u\in N_G[v]}f(u)\geq k$ for all $v\in V$. The \emph{signed $k$-domination number} of $G$ is the minimum value of $\sum_{v\in V}f(v)$ taken over all signed $k$-dominating functions of $G$. The \emph{signed total $k$-dominating function} and \emph{signed total $k$-domination number} of $G$ can be similarly defined by changing the closed neighborhood $N_G[v]$ to the open neighborhood $N_G(v)$ in the definition. The \emph{upper signed $k$-domination number} is the maximum value of $\sum_{v\in V}f(v)$ taken over all \emph{minimal} signed $k$-dominating functions of $G$. In this paper, we study these graph parameters from both algorithmic complexity and graph-theoretic perspectives. We prove that for every fixed $k\geq 1$, the problems of computing these three parameters are all \NP-hard. We also present sharp lower bounds on the signed $k$-domination number and signed total $k$-domination number for general graphs in terms of their minimum and maximum degrees, generalizing several known results about signed domination.Comment: Accepted by JCMC

Lower Bounds on Nonnegative Signed Domination Parameters in Graphs

Let $1 \leq k \leq n$ be a positive integer. A {\em nonnegative signed $k$-subdominating function} is a function $f:V(G) \rightarrow \{-1,1\}$ satisfying $\sum_{u\in N_G[v]}f(u) \geq 0$ for at least $k$ vertices $v$ of $G$. The value $\min\sum_{v\in V(G)} f(v)$, taking over all nonnegative signed $k$-subdominating functions $f$ of $G$, is called the {\em nonnegative signed $k$-subdomination number} of $G$ and denoted by $\gamma^{NN}_{ks}(G)$. When $k=|V(G)|$, $\gamma^{NN}_{ks}(G)=\gamma^{NN}_s(G)$ is the {\em nonnegative signed domination number}, introduced in \cite{HLFZ}. In this paper, we investigate several sharp lower bounds of $\gamma^{NN}_s(G)$, which extend some presented lower bounds on $\gamma^{NN}_s(G)$. We also initiate the study of the nonnegative signed $k$-subdomination number in graphs and establish some sharp lower bounds for $\gamma^{NN}_{ks}(G)$ in terms of order and the degree sequence of a graph $G$.Comment: 9 page

On Complexities of Minus Domination

A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x \in V. The minus-domination number \gamma^{-}(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1. In this paper we show that the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d. The minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs. It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameter algorithm for minus-domination. 79,1 5

New bounds on the signed domination numbers of graphs

In this paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we present a lower bound on signed domination number of trees in terms of the order, leaves and support vertices

On the Strong Roman Domination Number of Graphs

Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman domination in graphs was described. A Roman dominating function for a graph $G=(V,E)$, is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has at least a neighbor $w$ in $G$ for which $f(w)=2$. The Roman domination number of a graph is the minimum weight, $\sum_{v\in V}f(v)$, of a Roman dominating function. In this paper we initiate the study of a new parameter related to Roman domination, which we call strong Roman domination number and denote it by $\gamma_{StR}(G)$. We approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. In particular, we first show that the decision problem regarding the computation of the strong Roman domination number is NP-complete, even when restricted to bipartite graphs. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, we prove that for any tree $T$ of order $n\ge 3$, $\gamma_{StR}(T)\le 6n/7$ and characterize all extremal trees.Comment: 23 page

Bounds on the nonnegative signed domination number of graphs

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Tur\'{a}n we bound $\gamma^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. Graph Theory Appl. 4 (2) (2016), 231--237], simultaneously. Also, we bound $\gamma^{NN}_s(T)$ for a tree $T$ from above and below and characterize all trees attaining the bounds

Analogous to cliques for (m,n)-colored mixed graphs

Vertex coloring of a graph $G$ with $n$-colors can be equivalently thought to be a graph homomorphism (edge preserving vertex mapping) of $G$ to the complete graph $K_n$ of order $n$. So, in that sense, the chromatic number $\chi(G)$ of $G$ will be the order of the smallest complete graph to which $G$ admits a homomorphism to. As every graph, which is not a complete graph, admits a homomorphism to a smaller complete graph, we can redefine the chromatic number $\chi(G)$ of $G$ to be the order of the smallest graph to which $G$ admits a homomorphism to. Of course, such a smallest graph must be a complete graph as they are the only graphs with chromatic number equal to their order. The concept of vertex coloring can be generalize for other types of graphs. Naturally, the chromatic number is defined to be the order of the smallest graph (of the same type) to which a graph admits homomorphism to. The analogous notion of clique turns out to be the graphs with order equal to their (so defined) "chromatic number". These "cliques" turns out to be much more complicated than their undirected counterpart and are interesting objects of study. In this article, we mainly study different aspects of "cliques" for signed (graphs with positive or negative signs assigned to each edge) and switchable signed graphs (equivalence class of signed graph with respect to switching signs of edges incident to the same vertex).Comment: arXiv admin note: substantial text overlap with arXiv:1411.719

Complexity and Computation of Connected Zero Forcing

Zero forcing is an iterative graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. It is NP-hard to find a minimum zero forcing set - a smallest set of initially colored vertices which forces the entire graph to be colored. We show that the problem remains NP-hard when the initially colored set induces a connected subgraph. We also give structural results about the connected zero forcing sets of a graph related to the graph's density, separating sets, and certain induced subgraphs, and we characterize the cardinality of the minimum connected zero forcing sets of unicyclic graphs and variants of cactus and block graphs. Finally, we identify several families of graphs whose connected zero forcing sets define greedoids and matroids