Partial domination - the isolation number of a graph

We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed neighborhood of $D$. Furthermore, the bound is sharp. This seems to be the first result in the direction of partial domination with constrained structure on the graph induced by the non-dominated vertices, which we further elaborate in this paper.Comment: 28 page

Traceability of Connected Domination Critical Graphs

A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is connected. The connected domination number of $G$, $\gamma_c(G)$, is the minimum cardinality of a connected dominating set of $G$. A graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G)$ is equal to $k$ and $\gamma_{c}(G + uv) < k$ for every pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut-vertices of $G$. It is known that if $G$ is a $k$-$\gamma_{c}$-critical graph, then $G$ has at most $k - 2$ cut-vertices, that is $\zeta \le k - 2$. In this paper, for $k \ge 4$ and $0 \le \zeta \le k - 2$, we show that every $k$-$\gamma_{c}$-critical graph with $\zeta$ cut-vertices has a hamiltonian path if and only if $k - 3 \le \zeta \le k - 2$.Comment: 26 page

CONNECTED DOMINATION NUMBER OF A GRAPH AND ITS COMPLEMENT

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γc(G) is the minimum size of such a set. Let δ ∗ (G) = min{δ(G),δ(G)}, where G is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and G are both connected, γc(G) + γc(G) ≤ δ ∗ (G) + 4 − (γc(G) − 3)(γc(G) − 3). As a corollary, γc(G) + γc(G) ≤ 3n 4 when δ ∗ (G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that γc(G) + γc(G) ≤ δ ∗ (G) + 2 when γc(G),γc(G) ≥ 4. This bound is sharp when δ ∗ (G) = 6, and equality can only hold when δ ∗ (G) = 6. Finally, we prove that γc(G)γc(G) ≤ 2n − 4 when n ≥ 7, with equality only for paths and cycles