Connected [1,2]-Sets in Graphs

Abstract

A subset $S \subseteq V(G)$ in a graph $G=(V,E)$ is a connected [j, k]-set, if it satisfies that G[S] is a connected subgraph of G and every vertex $v \in V \setminus S$, $j \le |N(v) \cap S| \le k$ for non-negative integers $j<k$. In this paper, we study the connected [1, 2]-domination number of graph G, which denoted $\gamma _{c[1,2]}(G)$. We discussed the graphs with $\gamma _{c[1,2]}(G)=n$ and the graphs with $\gamma _{c[1,2]}(G)=\gamma _c(G)$. In the end, the CONNECTED [1, 2]-SET Problem has been proved to be NP-complete for bipartite graphs.

Supported by Natural science fundation of Zhejiang province (No. LY14F020040), Natural Science Foundation of China (No. 61173002)and Natural Science Foundation of Jiangsu University (No. 16KJB110011).