Structural and Polynomial-Time Results on Core and Corona in Odd-Bicyclic Graphs
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| Format: | Preprint |
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2026
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| _version_ | 1866910050270838784 |
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| author | Pereyra, Kevin |
| author_facet | Pereyra, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11420 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structural and Polynomial-Time Results on Core and Corona in Odd-Bicyclic Graphs Pereyra, Kevin Combinatorics 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. |
| title | Structural and Polynomial-Time Results on Core and Corona in Odd-Bicyclic Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.11420 |