Let $G=(V,E)$ be a simple graph. A $3$-valued function $fV\left(G\right)\to \{-1,0,1\}$ is said to be a minus dominating function if for every vertex $v\in V$, $f\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}f\left(u\right)\ge 1$, where $N\left[v\right]$ is the closed neighborhood of $v$. The weight of a minus dominating function $f$ on $G$ is $f\left(V\right)={\sum }_{v\in V}f\left(v\right)$. The minus domination number of a graph $G$, denoted by ${\gamma }^{-}\left(G\right)$, equals the minimum weight of a minus dominating function on $G$. In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$, then $${\gamma }^{-}\left(G\right)\ge 4\left(\sqrt{n+1}-1\right)-n.$$ (2) For any negative integer $k$ and any positive integer...

The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_f$ of all finite undirected graphs. If G is a graph from $C_f$, then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

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 $(0,2)$-graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0,\lambda )$-graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0,2)$-graph. $(0,2)$-graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0,\lambda )$-graphs and more specifically...

Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_G\left(x\right)\subseteq N_G\left[u\right]\cup N_G\left[v\right]$, where $N_G\left[x\right]=N_G\left(x\right)\cup x$. In the case where G is a claw-free graph, G* is equal to G². We define $\sigma °ₜ=min{\sum }_{x\in S}{d}_G\left(x\right):SisanindependentsetinG*and\left|S\right|=t$. We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

By a ternary structure we mean an ordered pair $(X_0,T_0)$, where $X_0$ is a finite nonempty set and $T_0$ is a ternary relation on $X_0$. By the underlying graph of a ternary structure $(X_0,T_0)$ we mean the (undirected) graph $G$ with the properties that $X_0$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if $$\{x\in X_0{T}_0(u,x,v)\}\cup \{x\in X_0{T}_0(v,x,u)\}=\{u,v\}.$$ A ternary structure $(X_0,T_0)$ is said to be the B-structure of a connected graph $G$ if $X_0$ is the vertex set of $G$ and the following statement holds for all $u,x,y\in X_0$: $T_0(x,u,y)$ if and only if $u$ belongs to an...

A graph $G$ is stratified if its vertex set is partitioned into classes, called strata. If there are $k$ strata, then $G$ is $k$-stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color $X$ in a stratified graph $G$, the $X$-eccentricity $e_X\left(v\right)$ of a vertex $v$ of $G$ is the distance between $v$ and an $X$-colored vertex furthest from $v$. The minimum $X$-eccentricity among the vertices of $G$ is the $X$-radius ${\mathrm{r}ad}_{X}G$ of $G$...

The additive stretch number $s_{add}\left(G\right)$ of a graph G is the maximum difference of the lengths of a longest induced path and a shortest induced path between two vertices of G that lie in the same component of G.We prove some properties of minimal forbidden configurations for the induced-hereditary classes of graphs G with $s_{add}\left(G\right)\le k$ for some k ∈ N₀ = 0,1,2,.... Furthermore, we derive characterizations of these classes for k = 1 and k = 2.

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...