Papers › On the finiteness of k-vertex-critical 2P₂-free graphs with forbidden induced squids or bulls
On the finiteness of k-vertex-critical 2P₂-free graphs with forbidden induced squids or bulls
Melvin Adekanye, Christopher Bury, Ben Cameron, Thaler Knodel
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
A graph is k-vertex-critical if χ(G)=k but χ(G-v)<k for all v∈V(G) and (G,H)-free if it contains no induced subgraph isomorphic to G or H. We show that there are only finitely many k-vertex-critical (2P₂,H)-free graphs for all k when H is isomorphic to any of the following graphs of order 5: bull, chair, claw+P₁, or diamond+P₁. The latter three are corollaries of more general results where H is isomorphic to (m, ℓ)-squid for m=3,4 and any ℓ≥1 where an (m,ℓ)-squid is the graph obtained from an m-cycle by attaching ℓ leaves to a single vertex of the cycle. For each of the graphs H above and any fixed k, our results imply the existence of polynomial-time certifying algorithms for deciding the k-colourability problem for (2P₂,H)-free graphs. Further, our structural classifications allow us to exhaustively generate, with aid of computer search, all k-vertex-critical (2P₂,H)-free graphs for k≤7 when H=bull or H=(4,1)-squid (also known as banner).
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections