A Linear Algorithm for Computing γ[1,2]-set in Generalized Series-Parallel Graphs

For a graph G = (V,E), a set S ⊆ V is a [1, 2]-set if it is a dominating set for G and each vertex v ∈ V \S is dominated by at most two vertices of S, i.e. 1 ≤ |N(v)∩S| ≤ 2. Moreover a set S ⊆ V is a total [1, 2]-set if for each vertex of V , it is the case that 1 ≤ |N(v)∩S| ≤ 2. The [1, 2]-domination number of G, denoted γ[1,2](G), is the minimum number of vertices in a [1, 2]-set. Every [1, 2]-set with cardinality of γ[1,2](G) is called a γ[1,2]-set. Total [1, 2]-domination number and γt[1,2]-sets of G are defined in a similar way. This paper presents a linear time algorithm to find a γ[1,2]-set and a γt[1,2]-set in generalized series-parallel graphs.

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