Construction of minimum edge-fault tolerant connected dominating set in a general graph

The Minimum connected dominating set problem (MinCDS) is a classical combinatorial optimization problem and has attached a lot of attention due to its application in wireless sensor networks (WSNs). Although the minimum k-connected m-fold dominating set problem (Min(k, m)-CDS), which takes vertex fault tolerance into consideration, has been extensively studied in recent years, studies on edge fault tolerant CDS only start very recently. In this paper, we study the edge analog of Min(k, m)-CDS, denoted as Min(k, m)-ECDS, which aims to find $$S\subseteq V(G)$$ S ⊆ V ( G ) such that the subgraph of G induced by S is k-edge connected and for any $$v\in V\setminus S$$ v ∈ V \ S , there are at least m edges between v and S. We give a greedy algorithm for Min(k, m)-ECDS on a general graph, with a theoretically guaranteed approximation ratio at most $$(2k-1)\ln \Delta +O(1)$$ ( 2 k - 1 ) ln Δ + O ( 1 ) , where $$\Delta $$ Δ is the maximum degree of G. When applied on an unit disk graph (UDG), the approximation ratio is at most $$10k-\frac{5}{k}+O(1)$$ 10 k - 5 k + O ( 1 ) when $$m\le 5$$ m ≤ 5 and $$14k+O(1)$$ 14 k + O ( 1 ) when $$m>5$$ m > 5 . In particular, our algorithm on Min(2, 2)-ECDS has approximation ratio at most 23.5, which improves the ratio 30.51 obtained in Liang et al. (Wirel Commun Mob Comput, 2021).