Two Kinds of Equicoverable Paths and Cycles

Abstract

Packing and covering are dual problems in graph theory. A graph $G$ is called $H$-equipackable if every maximal $H$-packing in $G$ is also a maximum $H$-packing in $G$. Dually, a graph $G$ is called $H$-equicoverable if every minimal $H$-covering in $G$ is also a minimum $H$-covering in $G$. In 2012, Zhang characterized two kinds of equipackable paths and cycles: $P_k$-equipackable paths and cycles, and $M_k$-equipackable paths and cycles. In this paper, we characterize $P_k$-equicoverable ($k>3$) paths and cycles, and $M_k$-equicoverable ($k>2$) paths and cycles.