Abstract
A matching
M in a graph
G is said to be extendable if there exists a perfect matching of
G containing
M. Also,
M is said to be a distance
d matching if the shortest distance between a pair of edges in
M is at least
d. A graph
G is distance
d matchable if every distance
d matching is extendable in
G, regardless of its size. In this paper, we study the class of distance
d matchable graphs. In particular, we prove that for every integer
k with
k
≥
3, there exists a positive integer
d such that every connected, locally
(k−1)‐connected
K1,k‐free graph of even order is distance
d matchable. We also prove that every connected, locally
k‐connected
K1,k‐free graph of even order is distance
3 matchable. Furthermore, we make more detailed analysis of
K1,4‐free graphs and study their distance matching extension properties.