COMBINATORIAL BOUNDS FOR DOUBLE DOMINATION INTEGRITY IN GRAPHS: A PLANAR REFINEMENT FOR NETWORK ROBUSTNESS

Vulnerability parameters in graphs aim to quantify the trade-off between an “attack cost” (removing vertices) and the resulting structural damage. Double domination integrity DDI(G) is an invariant that minimizes |S| + m(G − S) over double dominating sets S (closed neighborhoods), thereby combining redundancy in coverage with the size of the largest surviving component. General and computable bounds for DDI(G) in terms of basic invariants are still scarce. In this paper we develop a combinatorial framework based on an edge partition relative to a DDI-set and extremal estimates on the induced parts. This yields an explicit lower bound for all connected graphs in terms of the order n and the cyclomatic number β(G), and a sharper bound for connected planar graphs using the planar edge constraint; we also isolate the special situation where a planar graph admits a double dominating set of size 2. We discuss tightness on standard families (including complete and complete bipartite graphs) and illustrate how the bounds can be used as scalable surrogates for robustness when exact computation is infeasible.