The restrained domination number ${\gamma }^r\left(G\right)$ and the total restrained domination number ${\gamma }_t^r\left(G\right)$ of a graph $G$ were introduced recently by various authors as certain variants of the domination number $\gamma \left(G\right)$ of $\left(G\right)$. A well-known numerical invariant of a graph is the domatic number $d\left(G\right)$ which is in a certain way related (and may be called dual) to $\gamma \left(G\right)$. The paper tries to define analogous concepts also for the restrained domination and the total restrained domination and discusses the sense of such new definitions.

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r\left(v\right|W)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $dimG$. A set $S$ of vertices in $G$ is a dominating set...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by ${\gamma }_r\left(G\right)$, is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then ${\gamma }_r\left(U\right)\ge ⎡n/3⎤$, and provide a characterization of graphs achieving this bound.

A Roman dominating function on a graph G is a function f:V(G) → 0,1,2 satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f\left(V\left(G\right)\right)={\sum }_{u\in V\left(G\right)}f\left(u\right)$. The Roman domination number, ${\gamma }_R\left(G\right)$, of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_R\left(G\right)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E’ ⊆ E(G) for which ${\gamma }_R(G-E^{\text{'}})>{\gamma }_R\left(G\right)$....

Článek pojednává o matematických úlohách souvisejících se šachovnicí a šachovými figurami. Ze šachu však budeme potřebovat pouze pravidla pro pohyb figur po šachovnici. Postupně se zaměřujeme na jezdcovy procházky po obdélníkových šachovnicích a dále na tzv. nezávislost a dominanci figur a vztah mezi nimi na čtvercových šachovnicích. Ukážeme, že některé problémy lze řešit elegantněji, pokud je přeformulujeme v řeči teorie grafů.

A secure (total) dominating set of a graph G = (V,E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V-X, there exists x ∈ X adjacent to u such that $(X-x)\cup u$ is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number ${\gamma }_s\left(G\right)\left({\gamma }_{st}\left(G\right)\right)$. We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then ${\gamma }_{st}\left(G\right)\le {\gamma }_s\left(G\right)$. We also show that ${\gamma }_{st}\left(G\right)$ is at most twice the clique covering number of...

Consider a graph whose vertices play the role of members of the opposing groups. The edge between two vertices means that these vertices may defend or attack each other. At one time, any attacker may attack only one vertex. Similarly, any defender fights for itself or helps exactly one of its neighbours. If we have a set of defenders that can repel any attack, then we say that the set is secure. Moreover, it is strong if it is also prepared for a raid of one additional foe who can strike anywhere....

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number ${\gamma }_{\mathrm{s}}^2\left(G\right)$ and the signed total 2-domination number ${\gamma }_{\mathrm{st}}^2\left(G\right)$ of a graph $G$ are variants of the signed domination number ${\gamma }_{\mathrm{s}}\left(G\right)$ and the signed total domination number ${\gamma }_{\mathrm{st}}\left(G\right)$. Their values for caterpillars are studied.

The paper studies the signed domination number and the minus domination number of the complete bipartite graph $K_{p,q}$.

Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If ${\sum }_{x\in N¯\left[v\right]}f\left(x\right)\ge 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number ${\gamma }_S\left(D\right)$ of D. A set $f₁,f₂,...,f_d$ of signed dominating functions on D with the property that ${\sum }_{i=1}^d{f}_i\left(x\right)\le 1$ for each...

The concept of signed domination number of an undirected graph (introduced by J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and P. J. Slater) is transferred to directed graphs. Exact values are found for particular types of tournaments. It is proved that for digraphs with a directed Hamiltonian cycle the signed domination number may be arbitrarily small.

Let k ≥ 2 be an integer. A function f: V(G) → −1, 1 defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v] f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence...

Let k ≥ 1 be an integer, and G = (V, E) be a finite and simple graph. The closed neighborhood NG[e] of an edge e in a graph G is the set consisting of e and all edges having a common end-vertex with e. A signed Roman edge k-dominating function (SREkDF) on a graph G is a function f : E → {−1, 1, 2} satisfying the conditions that (i) for every edge e of G, ∑x∈NG[e] f(x) ≥ k and (ii) every edge e for which f(e) = −1 is adjacent to at least one edge e′ for which f(e′) = 2. The minimum of the values...

The signed total domination number of a graph is a certain variant of the domination number. If $v$ is a vertex of a graph $G$, then $N\left(v\right)$ is its oper neighbourhood, i.e. the set of all vertices adjacent to $v$ in $G$. A mapping $f:V\left(G\right)\to \{-1,1\}$, where $V\left(G\right)$ is the vertex set of $G$, is called a signed total dominating function (STDF) on $G$, if ${\sum }_{x\in N\left(v\right)}f\left(x\right)\ge 1$ for each $v\in V\left(G\right)$. The minimum of values ${\sum }_{x\in V\left(G\right)}f\left(x\right)$, taken over all STDF’s of $G$, is called the signed total domination number of $G$ and denoted by ${\gamma }_{\mathrm{s}t}\left(G\right)$. A theorem stating lower bounds for ${\gamma }_{\mathrm{s}t}\left(G\right)$ is stated for the...

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman dominating function (STRDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight of an STRDF f is w(f) = ∑v∈V (D) f(v). The signed total Roman domination number γstR(D) of D is the...

Define a complete subgraph Q to be simplicial in a graph G when Q is contained in exactly one maximal complete subgraph ('maxclique') of G; otherwise, Q is nonsimplicial. Several graph classes-including strong p-Helly graphs and strongly chordal graphs-are shown to have pairs of peculiarly related new characterizations: (i) for every k ≤ 2, a certain property holds for the complete subgraphs that are in k or more maxcliques of G, and (ii) in every induced subgraph H of G, that same property...