A graph $G$ is called an $S$-graph if its periphery $Peri\left(G\right)$ is equal to its center eccentric vertices $Cep\left(G\right)$. Further, a graph $G$ is called a $D$-graph if $Peri\left(G\right)\cap Cep\left(G\right)=\varnothing$. We describe $S$-graphs and $D$-graphs for small radius. Then, for a given graph $H$ and natural numbers $r\ge 2$, $n\ge 2$, we construct an $S$-graph of radius $r$ having $n$ central vertices and containing $H$ as an induced subgraph. We prove an analogous existence theorem for $D$-graphs, too. At the end, we give some properties of $S$-graphs and $D$-graphs.

The eccentricity $e\left(v\right)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$, and $u$ is an eccentric vertex for $v$ if its distance from $v$ is $d(u,v)=e\left(v\right)$. A vertex of maximum eccentricity in a graph $G$ is called peripheral, and the set of all such vertices is the peripherian, denoted $PeriG)$. We use $Cep\left(G\right)$ to denote the set of eccentric vertices of vertices in $C\left(G\right)$. A graph $G$ is called an S-graph if $Cep\left(G\right)=Peri\left(G\right)$. In this paper we characterize S-graphs with diameters less or equal to four, give some constructions of...

For a vertex $v$ of a connected oriented graph $D$ and an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices of $D$, the (directed distance) representation of $v$ with respect to $W$ is the ordered $k$-tuple $r\left(v\right|W)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k))$, where $d(v,w_i)$ is the directed distance from $v$ to $w_i$. The set $W$ is a resolving set for $D$ if every two distinct vertices of $D$ have distinct representations. The minimum cardinality of a resolving set for $D$ is the (directed distance) dimension $dim\left(D\right)$ of $D$. The dimension of a connected oriented graph need not be defined. Those oriented...

A subset $D$ of the vertex set $V\left(G\right)$ of a graph $G$ is called point-set dominating, if for each subset $S\subseteq V\left(G\right)-D$ there exists a vertex $v\in D$ such that the subgraph of $G$ induced by $S\cup \left\{v\right\}$ is connected. The maximum number of classes of a partition of $V\left(G\right)$, all of whose classes are point-set dominating sets, is the point-set domatic number $d_p\left(G\right)$ of $G$. Its basic properties are studied in the paper.

As introduced by F. Harary in 1994, a graph $G$ is said to be an $integral$ $sum$ $graph$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$, $uv$ is an edge of $G$ if and only if $f\left(u\right)+f\left(v\right)=f\left(w\right)$ for some vertex $w$ in $G$. We prove that every integral sum graph with a saturated vertex, except the complete graph $K_3$, has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be if it is adjacent to every...