相关论文: Further results on \([k]\)-Roman domination on cyl…

Different types of domination on the Sierpi\'nski graphs S(K_n,t) will be studied in this paper. More precisely, we propose a minimal dominating set for S(K_n,t) so that the exact values of their domination numbers, Roman domination…

We present an algorithm to compute the domination polynomial of the $m \times n$ grid, cylinder, and torus graphs and the king graph. The time complexity of the algorithm is $O(m^2n^2 \lambda^{2m})$ for the torus and $O(m^3n^2\lambda^m)$…

We provide two algorithms counting the number of minimum Roman dominating functions of a graph on n vertices in O(1.5673^n) time and polynomial space. We also show that the time complexity can be reduced to O(1.5014^n) if exponential space…

A Roman dominating function (RD-function) on a graph $G = (V(G), E(G))$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight $f(V(G))$ of a RD-function $f$ on $G$…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G=(V,E)$ if for every vertex $v\in V$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating…

A vertex subset S of a graph G is said to 2-dominate the graph if each vertex not in S has at least two neighbors in it. As usual, the associated parameter is the minimum cardinal of a 2-dominating set, which is called the 2-domination…

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We…

For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

For a graph $G= (V, E)$, a Roman dominating function is a map $f : V \rightarrow \{0, 1, 2\}$ satisfies the property that if $f(v) = 0$, then $v$ must have adjacent to at least one vertex $u$ such that $f(u)= 2$. The weight of a Roman…

For any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least…

We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, a double Roman dominating function $f$ is called a restrained double Roman dominating function (RDRD function) if the subgraph…

A Roman dominating function for a (non-weighted) graph $G=(V,E)$, is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $u\in V$ with $f(u)=0$ has at least {one} neighbor $v\in V$ such that $f(v)=2$. The minimum weight $\sum_{v\in…

Let $G=(V,E)$ be a simple graph of order $n$. A Majority Roman Dominating Function (MRDF) on a graph G is a function $f: V\rightarrow\{-1, +1, 2\}$ if the sum of its function values over at least half the closed neighborhoods is at least…

We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq…

Let $k$ be a positive integer. A {\em Roman $k$-dominating function} on a graph $G$ is a labeling $f:V (G)\longrightarrow \{0, 1, 2\}$ such that every vertex with label 0 has at least $k$ neighbors with label 2. A set…

For a positive integer $k$, a $\{k\}$-Roman dominating function of a graph $G = (V,E)$ is a function $f\colon V \rightarrow \{0,1,\ldots,k\}$ satisfying $f (N(v)) \geq k$ for each vertex $v\in V$ with $f (v) = 0$. Every graph $G$ satisfies…

The study of Roman domination has evolved to encompass a variety of challenging extensions, each contributing to the broader understanding of domination problems in graph theory. This paper explores the Perfect Location Signed Roman…

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…