Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by ${\gamma }_{tc}\left(G\right)$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then ${\gamma }_{tc}\left(T\right)\ge ⎡2n/3⎤$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.

Let $G$ be a finite connected graph with minimum degree $\delta$. The leaf number $L\left(G\right)$ of $G$ is defined as the maximum number of leaf vertices contained in a spanning tree of $G$. We prove that if $\delta \ge \frac{1}{2}(L\left(G\right)+1)$, then $G$ is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if $\delta \ge \frac{1}{2}\left(L\right(G)+1)$, then $G$ contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin....

A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The minimum number of vertices in a connected dominating set of $G$ is called the connected domination number of $G$, and is denoted by ${\gamma }_c\left(G\right)$. Let $G$ be a spanning subgraph of $K_{s,s}$ and let $H$ be the complement of $G$ relative to $K_{s,s}$; that is, $K_{s,s}=G\oplus H$ is a factorization of $K_{s,s}$. The graph $G$ is $k$-${\gamma }_c$-critical relative to $K_{s,s}$ if ${\gamma }_c\left(G\right)=k$ and ${\gamma }_c(G+e)<k$ for each edge $e\in E\left(H\right)$. First, we discuss some classes of graphs whether they are ${\gamma }_c$-critical...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the 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 for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $dim\left(G\right)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected...