Related papers: Roman Census: Enumerating and Counting Roman Domin…
In this paper, we study the dominating set problem in \emph{RDV graphs}, a graph class that lies between interval graphs and chordal graphs and is defined as the \textbf{v}ertex-intersection graphs of \textbf{d}ownward paths in a…
The \textsc{Dominating Set} problem is a classical and extensively studied topic in graph theory and theoretical computer science. In this paper, we examine the algorithmic complexity of several well-known exact-distance variants of…
This work is related to the extension of the well-known problem of Roman domination in graph theory to fuzzy graphs. A variety of approaches have been used to explore the concept of domination in fuzzy graphs. This study uses the concept of…
Given a graph $G$ with vertex set $V(G)$, a mapping $h : V(G) \rightarrow \lbrace 0, 1, 2, 3, 4, 5 \rbrace$ is called a quadruple Roman dominating function (4RDF) for $G$ if it holds the following. Every vertex $x$ such that $h(x)\in…
Let $\gamma(G)$ and $\gamma_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that $\gamma_t(G) \leq 2\gamma(G)$. We provide a characterization of a…
Finding Minimal Unsatisfiable Subsets (MUSes) of binary constraints is a common problem in infeasibility analysis of over-constrained systems. However, because of the exponential search space of the problem, enumerating MUSes is extremely…
It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In…
A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Because the minimum dominating set problem for unit…
Given a positive integer $k$, a $k$-dominating set in a graph $G$ is a set of vertices such that every vertex not in the set has at least $k$ neighbors in the set. A total $k$-dominating set, also known as a $k$-tuple total dominating set,…
Dense subgraphs of sparse graphs (communities), which appear in most real-world complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between…
A Roman $\{2\}$-dominating function (R2F) is a function $f:V\rightarrow \{0,1,2\}$ with the property that for every vertex $v\in V$ with $f(v)=0$ there is a neighbor $u$ of $v$ with $f(u)=2$, or there are two neighbors $x,y$ of $v$ with…
One of the well-known measurements of vulnerability in graph theory is domination. There are many kinds of dominating and relative types of sets in graphs. However, we are going to focus on Roman domination, which is a type of domination…
The problems of determining the minimum-sized \emph{identifying}, \emph{locating-dominating} and \emph{open locating-dominating codes} of an input graph are special search problems that are challenging from both theoretical and…
Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a…
Say that an edge of a graph $G$ dominates itself and every other edge adjacent to it. An edge dominating set of a graph $G=(V,E)$ is a subset of edges $E' \subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is…
Enumerating minimal transversals in a hypergraph is a notoriously hard problem. It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph. We provide…
A maximal double Roman dominating function (MDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ such that \textrm{(i) }every vertex $v$ with $f(v)=0$ is adjacent to least two vertices { assigned $2$ or to at least one…
We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating…
In this paper we deal with the calculation of the signed (total) Roman domination numbers, $\gamma_{sR}$ and $\gamma_{stR}$ respectively, on a few classes of planar graphs from the literature. We give proofs for the exact values of the…
Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS$_o$) algorithm to solve minimum dominating set problem in large…