Algorithms for the Maximum Edge Open Packing Problem

Abstract

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device can participate in at most one communication at a time and simultaneous links must avoid interference we consider a generalization of induced matching known as \emph{edge open packing}. Two edges of a graph are said to conflict if a third edge connects one endpoint of each; an \emph{edge open packing set} is a set of edges containing no such conflicting pair. The largest cardinality of such a set is the \emph{edge open packing number} of a graph. In this work, we study the computational complexity of the Maximum Edge Open Packing Problem. We give a polynomial-time algorithm for the problem in \emph{distance-hereditary graphs}, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on \emph{biconvex bipartite graphs}, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for \emph{chordal graphs}, parameterized by the clique number $ω$, running in $O(2^ω\cdot\mathrm{poly}(n))$ time.