Search

For a simple, undirected, connected graph G , a function h : V (G) → { 0 , 1 , 2 } which satisfies the following conditions is called a weakly connected Roman dominating function (WCRDF) of G with weight h (V) = ∑ p ∈ V h (p). (C1). For all q ∈ V with h (q) = 0 there exists a vertex r such that (q , r) ∈ E and h (r) = 2 and (C2). The graph with vertex set V (G) and edge set { (p , z) : h (p) ≥ 1 or h (z) ≥ 1 or both } is connected. The problem of determining WCRDF of minimum weight is called minimum weakly connected Roman domination problem (MWCRDP). In this paper, we show that MWCRDP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a 2 (1 +) (1 + ln (Δ − 1)) -approximation algorithm for the MWCRDP and show that the same cannot have (1 − δ) ln | V | ratio approximation algorithm for any δ > 0 unless P = N P. Next, we show that MWCRDP is APX-hard for graphs with Δ = 4. We also show that the domination and weakly connected Roman domination problems are not equivalent in computational complexity aspects. Finally, two different integer linear programming formulations for MWCRDP are proposed. [ABSTRACT FROM AUTHOR]

For a simple, undirected, connected graph G , a function h : V (G) → { 0 , 1 , 2 , 3 } which satisfies the following conditions is called a total Roman {3}-dominating function (TR3DF) of G with weight h (V) = ∑ p ∈ V h (p) : (C1) For every vertex u ∈ V if h (u) = 0 , then u has m (m ≥ 1) neighbors such that whose sum is at least 3, and if h (u) = 1 , then u has n (n ≥ 1) neighbors such that whose sum is at least 2. (C2) The subgraph induced by the set of vertices labeled one, two or three has no isolated vertices. For a graph G , the smallest possible weight of a TR3DF of G denoted γ t { R 3 } (G) is known as the total Roman { 3 } -domination number of G. The problem of determining γ t { R 3 } (G) of a graph G is called minimum total Roman {3}-domination problem (MTR3DP). In this paper, we show that the problem of deciding if G has a TR3DF of weight at most l for chordal graphs is NP-complete. We also show that MTR3DP is polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a 3 (ln (Δ − 0. 5) + 1. 5) -approximation algorithm for the MTR3DP and show that the same cannot have (1 − δ) ln | V | ratio approximation algorithm for any δ > 0 unless NP ⊆ DTIME (| V | O (log log | V |)). Next, we show that MTR3DP is APX-complete for graphs with Δ = 4. We also show that the domination and total Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MTR3DP. [ABSTRACT FROM AUTHOR]

Let G be a simple, undirected graph. A function g : V(G) → {0, 1, 2, 3} having the property that ∑v∈NG(u)g(v)≥3, if g(u) = 0, and ∑v∈NG(u)g(v)≥2, if g(u) = 1 for any vertex u ∈ G, where NG(u) is the set of vertices adjacent to u in G, is called a Roman {3}-dominating function (R3DF) of G. The weight of a R3DF g is the sum g(V)=∑v∈Vg(v). The minimum weight of a R3DF is called the Roman {3}-domination number and is denoted by γ{R3}(G). Given a graph G and a positive integer k, the Roman {3}-domination problem (R3DP) is to check whether G has a R3DF of weight at most k. In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. The minimum Roman {3}-domination problem (MR3DP) is to find a R3DF of minimum weight in the input graph. We show that MR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. We propose a 3(1 + ln(Δ − 1))-approximation algorithm for the MR3DP, where Δ is the maximum degree of G and show that the MR3DP problem cannot be approximated within (1 − ε)ln|V| for any ε > 0 unless NP ⊆ DTIME(|V|O(loglog|V|)). Next, we show that the MR3DP problem is APX-complete for graphs with maximum degree 4. We also show that the domination and Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, an ILP formulation for MR3DP is proposed. [ABSTRACT FROM AUTHOR]

A set D ⊆ V is called a total dominating set of a graph G = ( V , E ) , if for all v ∈ V , | N G ( v ) ∩ D | ≥ 1 . A total dominating set D of G = ( V , E ) is called a total outer-connected dominating set if G [ V ∖ D ] is connected. The Minimum Total Outer-connected Domination problem for a graph G = ( V , E ) is to find a total outer-connected dominating set of minimum cardinality. The Total Outer-connected Domination Decision problem, the decision version of the Minimum Total Outer-connected Domination problem, is known to be NP-complete for general graphs. In this paper, we strengthen this NP-completeness result by showing that the Total Outer-connected Domination Decision problem remains NP-complete for chordal graphs, split graphs, and doubly chordal graphs. We prove that the Total Outer-connected Domination Decision problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for computing the total outer-connected domination number, the cardinality of a minimum total outer-connected dominating set, of a tree. We prove that the Minimum Total Outer-connected Domination problem cannot be approximated within a factor of ( 1 − ε ) ln | V | for any ε > 0 , unless NP ⊆ DTIME ( | V | O ( log log | V | ) ) . Finally, we show that the Minimum Total Outer-connected Domination problem is APX-complete for bounded degree graphs. [ABSTRACT FROM AUTHOR]