Structural and Polynomial-Time Results on Core and Corona in Odd-Bicyclic Graphs

Abstract

Let $\core G$ and $\corona G$ denote the intersection and the union, respectively, of all maximum independent sets of a graph $G$. In this work, we show that for a graph with at most two odd cycles, $\a{\core G}+\a{\corona G}$ is equal to $2\alpha(G)$, $2\alpha(G)+1$, or $2\alpha(G)+2$, and we precisely characterize when each value occurs. We further characterize graphs with at most two odd cycles that admit the core--corona partition $V(G)=\corona G\ud N(\core G)$, extending known results for K\"onig--Egerv\'ary and almost bipartite graphs. Deciding whether $\core G=\emptyset$ is known to be \textbf{NP}-hard. As an algorithmic consequence of the obtained results, we show that the core, independence number and the corona can be computed in polynomial time for this class of graphs.