TY - JOUR

T1 - Transversals of longest paths

AU - Cerioli, Márcia R.

AU - Fernandes, Cristina G.

AU - Gómez, Renzo

AU - Gutiérrez, Juan

AU - Lima, Paloma T.

N1 - Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2020/3

Y1 - 2020/3

N2 - Let lpt(G) be the minimum cardinality of a transversal of longest paths in G, that is, a set of vertices that intersects all longest paths in a graph G. There are several results in the literature bounding the value of lpt(G) in general or in specific classes of graphs. For instance, lpt(G)=1 if G is a connected partial 2-tree, and a connected partial 3-tree G is known with lpt(G)=2. We prove that lpt(G)≤3 for every connected partial 3-tree G; that lpt(G)≤2 for every planar 3-tree G; and that lpt(G)=1 if G is a connected bipartite permutation graph or a connected full substar graph. Our first two results can be adapted for broader classes, improving slightly some known general results: we prove that lpt(G)≤k for every connected partial k-tree G and that lpt(G)≤max{1,ω(G)−2} for every connected chordal graph G, where ω(G) is the cardinality of a maximum clique in G.

AB - Let lpt(G) be the minimum cardinality of a transversal of longest paths in G, that is, a set of vertices that intersects all longest paths in a graph G. There are several results in the literature bounding the value of lpt(G) in general or in specific classes of graphs. For instance, lpt(G)=1 if G is a connected partial 2-tree, and a connected partial 3-tree G is known with lpt(G)=2. We prove that lpt(G)≤3 for every connected partial 3-tree G; that lpt(G)≤2 for every planar 3-tree G; and that lpt(G)=1 if G is a connected bipartite permutation graph or a connected full substar graph. Our first two results can be adapted for broader classes, improving slightly some known general results: we prove that lpt(G)≤k for every connected partial k-tree G and that lpt(G)≤max{1,ω(G)−2} for every connected chordal graph G, where ω(G) is the cardinality of a maximum clique in G.

KW - Bipartite permutation

KW - Full substar graph

KW - Longest path

KW - Planar 3-tree

KW - Transversal

UR - http://www.scopus.com/inward/record.url?scp=85075155341&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2019.111717

DO - 10.1016/j.disc.2019.111717

M3 - Article

AN - SCOPUS:85075155341

SN - 0012-365X

VL - 343

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 3

M1 - 111717

ER -