Perfect weighted divisibility is equivalent to perfect divisibility

Abstract

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\ddot{I}‰(H[B])<\ddot{I}‰(H)$. A graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. In this paper, we prove that the perfect divisibility of a graph is equivalent to its perfect weighted divisibility.

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