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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]

Zou et al. (2019) proposed an algorithm for calculating a minimum secure dominating set of a proper interval graph. However, we find a counterexample for which the algorithm outputs an incorrect subset of vertices. In this paper, we provide a modified algorithm and the proofs for the correctness. [ABSTRACT FROM AUTHOR]

A set S ⊂ V is a co-secure dominating set of a graph G = (V , E) if S is a dominating set, and for each u ∈ S there exists a vertex v ∈ V ∖ S such that u v ∈ E and (S ∖ { u }) ∪ { v } is a dominating set. Note that | V | > 1. The minimum cardinality of a co-secure dominating set in G is the co-secure domination number of G. In this paper, we propose a linear-time algorithm for finding the co-secure domination number of proper interval graphs. [ABSTRACT FROM AUTHOR]