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 $2α(G)$, $2α(G)+1$, or $2α(G)+2$, 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 König--Egerváry 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.

Theorems & Definitions (32)

• Theorem 3.1: larson2011critical
• Theorem 3.2: pulleyblank1979minimum
• Theorem 3.4: kevincorecorona2biciclicbicritical
• Theorem 3.5: levit2012critical
• Theorem 3.6: Butenko & Trukhanovbutenko2007using
• Lemma 3.7
• proof
• Theorem 3.8: jarden2019monotonic
• Theorem 3.9
• Corollary 3.10