Maximum Edge Open Packing on AT-Free, Chordal, and Convex Bipartite Graphs

Abstract

An edge open packing is a set of edges whose endpoints induce a disjoint union of stars. We study the problem of finding a maximum edge open packing in AT-free, chordal, and convex bipartite graphs. For AT-free graphs, we use the oriented star-conflict graph $A_G$, introduced by Das and Santra, for which ${\rho }_e^o(G)=\alpha (A_G)$, and prove that $A_G$ is AT-free whenever $G$ is AT-free. Consequently, a known maximum independent set algorithm for AT-free graphs yields an $O(n^2+m^4)$-time algorithm for Maximum Edge Open Packing, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively. For chordal graphs, we develop an $O(n^4)$-time dynamic programming algorithm over a nice tree decomposition derived from a clique tree, exploiting the structural fact that the endpoint set of an edge open packing intersects every clique in at most two vertices. Finally, for convex bipartite graphs, we obtain an $O(n^4)$-time dynamic programming algorithm based on two boundary indices that separate consecutive star components. This result extends the previously known algorithm for biconvex bipartite graphs and, in particular, provides an improved explicit running-time bound for that class.