On perfect and unique maximum independent sets in graphs

Volkmann, Lutz

Mathematica Bohemica, Tome 129 (2004) no. 3, pp. 273-282.

Résumé

A perfect independent set $I$ of a graph $G$ is defined to be an independent set with the property that any vertex not in $I$ has at least two neighbors in $I$. For a nonnegative integer $k$, a subset $I$ of the vertex set $V(G)$ of a graph $G$ is said to be $k$-independent, if $I$ is independent and every independent subset $I^{\prime }$ of $G$ with $|I^{\prime }|\ge |I|-(k-1)$ is a subset of $I$. A set $I$ of vertices of $G$ is a super $k$-independent set of $G$ if $I$ is $k$-independent in the graph $G[I,V(G)-I]$, where $G[I,V(G)-I]$ is the bipartite graph obtained from $G$ by deleting all edges which are not incident with vertices of $I$. It is easy to see that a set $I$ is $0$-independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of $G$. In this paper we mainly investigate connections between perfect independent sets and $k$-independent as well as super $k$-independent sets for $k=0$ and $k=1$.

Détail

DOI : 10.21136/MB.2004.134148

Classification : 05C69, 05C70
Keywords: independent sets; perfect independent sets; unique independent sets; strong unique independent sets; super unique independent sets