Some properties of generalized complements of a graph

Abstract

Let $P=\{V_{1},V_{2},\cdots,V_{k}\}$ be a partition of vertex set $V$ of $G$. The $k-$complement of $G$ denoted by $G_{k}^{P}$ is defined as follows: for all $V_{i}$ and $V_{j}$ in $P$, $i\neq j$, remove the edges between $V_{i}$ and $V_{j}$ and add edges between $V_{i}$ and $V_{j}$ which are not in $G$. The graph $G$ is k-self complementary with respect to $P$ if $G_{k}^{P}\cong G$. The k(i)-complement $G_{k(i)}^{P}$ of a graph $G$ with respect to $P$ is defined as follows: for all $V_{r}\in P$, remove edges inside $V_{r}$ and add edges which are not in $V_{r}$. In this paper we provide sufficient conditions for $G_{k}^{P}$ and $G_{k(i)}^{P}$ to be disconnected, regular, line preserving and Eulerian.