Mathematics and Computing for Scientific Innovation Vol. 1

Edge domination extends classical domination concepts to the edge set of a graph. In this study, an edge domatic decomposition is considered as a decomposition G1,G2, . . . ,Gn in which each connected subgraph Gi has edge domatic number i, and the maximum number of such subgraphs is termed the edge domatic decomposition number. The work examines three ladder-type graph families: the ladder graph Lp, the circular ladder graph CLp, and the triangular ladder graph T Lp. The analysis proceeds by constructing pairwise edge-disjoint edge dominating sets to determine the edge domatic numbers required for subsequent decompositions. For Lp, the edge domatic number is 3 for p ≥ 3, and the edge domatic decomposition number is also 3. For CLp, the edge domatic number is 3 when p = 3k + 2 and 4 otherwise, while the edge domatic decomposition number is 3 for p ≥ 3. The auxiliary graph Cp ⊙ K1 has edge domatic number 3 for p ≥ 3. For T Lp, the edge domatic number equals 3 when p = 2 and 4 when p > 2. Its edge domatic decomposition number is 2 for p = 2, 3 for p = 3, and 4 for p > 3. These results provide explicit edge domatic decompositions for the ladder graph families considered.