Geodesic
Non-empty intersection of longest paths in \(H\)-free graphs
James A. Long Jr.1 ; Kevin G. Milans1 ; Andrea Munaro2
1West Virginia University
2Queen's University Belfast
The electronic journal of combinatorics, Tome 30 (2023) no. 1
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We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$, and its independence number $\alpha(G)$ satisfies $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.
DOI : 10.37236/11277
Classification : 05C38, 05C12
Mots-clés : connected \(H\)-free graph, longest path transversal

James A. Long Jr.  1   ; Kevin G. Milans  1   ; Andrea Munaro  2

1 West Virginia University
2 Queen's University Belfast 
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     title = {Non-empty intersection of longest paths in {\(H\)-free} graphs},
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James A. Long Jr.; Kevin G. Milans; Andrea Munaro. Non-empty intersection of longest paths in \(H\)-free graphs. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11277

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