AbstractA bipartite graph $$G=(U,V,E)$$
G
=
(
U
,
V
,
E
)
is convex if the vertices in V can be linearly ordered such that for each vertex $$u\in U$$
u
∈
U
, the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in $$O(n+m)$$
O
(
n
+
m
)
time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex $$u\in U$$
u
∈
U
the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in $$O(n+m)$$
O
(
n
+
m
)
time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of $$O(n^2)$$
O
(
n
2
)
(Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. 10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.
Connectivity keeping paths in k-connected bipartite graphs
