The Independent Bondage Number of Planar Graphs Without Short Cycles

An independent dominating set I of a graph G is an independent set such that every vertex in V(G)−I has at least one neighbor in I. The independent domination number of G is the minimum cardinality of an independent dominating set of G. The independent bondage number of a graph G, denoted by bi(G), is the minimum cardinality of an edge set whose removal increases the independent domination number of G. In 2025, Gamlath et al. proved that for planar graphs G with δ(G)≥2, the following holds: bi(G)≤5 if G does not contain cycles of lengths 3 and 4, and bi(G)≤4 if G does not contain cycles of lengths 3, 4, 5, and 6. This motivates us to investigate the independent bondage number of planar graphs with δ(G)≥2 without cycles of lengths 4 and 5, or cycles of lengths 4, 5, 6, and 7. In this paper, we prove the following bounds for planar graphs G with δ(G)≥2: bi(G)≤5 if G does not contain cycles of lengths 4 and 5, and bi(G)≤4 if G does not contain cycles of lengths 4, 5, 6, and 7.