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In a graph, G = (V , E) without an isolated vertex, a dominating set D ⊆ V , is called a semitotal dominating set if for every vertex u ∈ D there is another vertex v ∈ D such that distance between u and v is at most two in G. Given a graph G = (V , E) without an isolated vertex, the Minimum Semitotal Domination problem is to find a minimum cardinality semitotal dominating set of G. The semitotal domination number , denoted by γ t 2 (G) , is the minimum cardinality of a semitotal dominating set of G. It is known that the decision version of the problem remains NP-complete even when restricted to chordal graphs, chordal bipartite graphs, and planar graphs. Galby et al. (2020) proved that the problem can be solved in polynomial time for bounded MIM-width graphs, which include many well known graph classes, but left the complexity of the problem in strongly chordal graphs unresolved. Henning and Pandey (2019) also asked to resolve the complexity status of the problem in strongly chordal graphs. In this paper, we resolve the complexity of the problem in strongly chordal graphs by designing a linear-time algorithm for the problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the complexity of the edge monitoring problem. A vertex v monitors an edge e if both endpoints together with v form a triangle in the graph. Given a graph G = (V , E) and a weight function on edges c where c (e) is the number of monitors that needs the edge e , the problem is to seek a minimum subset of monitors S such that every edge e in the graph is monitored by at least c (e) vertices in S. Therefore, we study the edge monitoring problem on several graph classes such as complete graphs, block graphs, cographs, split graphs, interval graphs and planar graphs. We also generalize the problem by adding weights on vertices. [ABSTRACT FROM AUTHOR]