Paired Domination in Cubic Bipartite Graphs

Source abstract

A paired dominating set of a graph $G$ is a dominating set $D$ such that $G[D]$ has a perfect matching. The minimum size of such a set is the paired domination number $\gpr(G)$. Desormeaux and Henning conjectured that every cubic bipartite graph $G$ of order $n$ satisfies $\gpr(G)\le n/2$. We prove the conjecture in the sharp integer form $\gpr(G)\le 2\lfloor |V(G)|/4\rfloor$ for every finite simple cubic bipartite graph $G$. The proof combines a directed contraction along a perfect matching, switching arguments based on dominator trees, a four-symbol boundary calculus for two-edge cuts, and the Gallai--Edmonds decomposition. Equality is attained by $K_{3,3}$ when $|V(G)|\equiv2\pmod4$ and by the cube $Q_3$ when $|V(G)|\equiv0\pmod4$.