Coronas and domination subdivision number of a graph

In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G \circ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.Comment: 9 pages, 4 figure

Total domination multisubdivision number of a graph

The domination multisubdivision number of a nonempty graph $G$ was defined as the minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to increase the domination number of $G$. Similarly we define the total domination multisubdivision number msd$_{\gamma_t}(G)$ of a graph $G$ and we show that for any connected graph $G$ of order at least two, msd$_{\gamma_t}(G)\leq 3.$ We show that for trees the total domination multisubdivision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees $T$ with msd$_{\gamma_t}(T)=1$.Comment: 15 pages, 1 figure, 9 reference

Critical graphs upon multiple edge subdivision

A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $\gamma(G)$ be the minimum cardinality among all dominating sets in $G.$ A graph $G$ is $\gamma$-$q$-{\it critical} if the smallest subset of edges whose subdivision necessarily increases $\gamma(G)$ has cardinality $q.$ In this paper we consider mainly $\gamma$-$q$-critical trees and give some general properties of $gamma$-$q$-critical graphs. In particular, we show that if $T$ is a $\gamma$-$q$-critical tree, then $1 \leq q \leq n(T)-1$ and we characterize extremal trees when $q=n(T)-1.$ Since a subdivision number {of a tree $T$} ${\rm sd}(T)$ is always $1,2$ or $3,$ we also characterize $\gamma$-2-critical trees $T$ with ${\rm sd}(T)=2$ and $\gamma$-3-critical trees $T$ with ${\rm sd}(T)=3.$Comment: 11 pages, 1 figur