Algorithmic Aspects of Quasi-Total Roman Domination in Graphs

For a simple, undirected graph $$G = (V, E)$$ , a Roman dominating function (RDF) $$f{:}V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that, every vertex u with $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) = 2$$ . The weight of a RDF is the sum $$f(V) = \sum _{v \in V}f(v)$$ . The minimum weight of a RDF is called the Roman domination number and is denoted by $$\gamma _{R}(G)$$ . Given a graph G and a positive integer k, the Roman domination problem (RDP) is to check whether G has a RDF of weight at most k. The RDP is known to be NP-complete for bipartite graphs. We strengthen this result by showing that this problem remains NP-complete for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. We show that $$\gamma _{R}(G)$$ is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph. We show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within $$(1 - \epsilon ) \ln |V|$$ for any $$\epsilon > 0$$ unless $$NP \subseteq DTIME(|V|^{O(\log \log |V|)})$$ and also propose a $$2(1+\ln (\varDelta +1))$$ -approximation algorithm for the MRDP, where $$\varDelta $$ is the maximum degree of G. Finally, we show that the MRDP is APX-complete for graphs with maximum degree 5.

For a simple, undirected graph $G(V,E)$, a function $h : V(G) rightarrow lbrace 0, 1, 2rbrace$ such that each edge $ (u,v)$ of $G$ is either incident with a vertex with weight at least one or there exists a vertex $w$ such that either $(u,w) in E(G)$ or $(v,w) in E(G)$ and $h(w) = 2$, is called a vertex-edge Roman dominating function (ve-RDF) of $G$. For a graph $G$, the smallest possible weight of a ve-RDF of $G$ which is denoted by $gamma_{veR}(G)$, is known as the textit{vertex-edge Roman domination number} of $G$. The problem of determining $gamma_{veR}(G)$ of a graph $G$ is called minimum vertex-edge Roman domination problem (MVERDP). In this article, we show that the problem of deciding if $G$ has a ve-RDF of weight at most $l$ for star convex bipartite graphs, comb convex bipartite graphs, chordal graphs and planar graphs is NP-complete. On the positive side, we show that MVERDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, a 2-approximation algorithm for MVERDP is presented. It is also shown that vertex cover and vertex-edge Roman domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MVERDP is presented.

For a simple, undirected graph $$G = (V,$$ E), an independent Roman dominating function (IRDF) $$f : V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that every vertex $$v \in V$$ with $$f(v) = 0$$ is adjacent to at least one vertex u, with $$f(u) = 2$$ , and no two vertices assigned positive values are adjacent. An independent Roman $$\lbrace 2 \rbrace $$ -dominating function (IR2DF) is an IRDF f with an additional property that, for every vertex $$v \in V$$ with $$f(v) = 0$$ , there exists at least two vertices $$x, y \in N_G(v)$$ with $$f(x) = f(y) = 1$$ . An independent double Roman dominating function (IDRDF) on G is a function $$f : V \rightarrow \lbrace 0, 1, 2, 3 \rbrace $$ , such that for every vertex $$v \in V$$ , if $$f(v) = 0$$ , then v has at least two neighbors x, y with $$f(x) = f(y) = 2$$ or one neighbor w with $$f(w) = 3$$ , and if $$f(v) = 1$$ , then v must have at least one neighbor w with $$f(w) \ge 2$$ , and no two vertices assigned positive values are adjacent. The weight of an IRDF (IR2DF, IDRDF) f is the sum $$f(V) = \sum _{v \in V}f(v)$$ . Given a graph G and a positive integer k, the independent Roman domination problem (IRDP), independent Roman $$\lbrace 2 \rbrace $$ -domination problem (IR2DP) and independent double Roman domination problem (IDRDP), respectively, is to check whether G has an IRDF, IR2DF, and IDRDF of weight at most k. The IRDP, IR2DP and IDRDP are known to be NP-complete for bipartite graphs. We investigate the complexity of these problems for dually chordal graphs and some subclasses of bipartite graphs, namely, star convex bipartite graphs and comb convex bipartite graphs. We also investigate the complexity of IR2DP and IDRDP for chordal graphs. The minimum independent Roman domination problem (MIRDP), minimum independent Roman $$\lbrace 2 \rbrace $$ -domination problem (MIR2DP), and minimum independent double Roman domination problem (MIDRDP), respectively, are to find an IRDF, IR2DF, and IDRDF of minimum weight in the input graph. We show that MIRDP, MIR2DP, and MIDRDP are linear time solvable for bounded tree-width graphs, chain graphs, and threshold graphs, a subclass of split graphs. We also show that the MIRDP, MIR2DP, and MIDRDP are APX-hard for graphs with bounded degree 4. Finally, we show that the domination problem and IRDP (IR2DP, IDRDP) are not equivalent in computational complexity aspects.

For a simple, undirected graph a Roman {2}-dominating function (R2DF) has the property that for every vertex with f(v) = 0, either there exists a vertex with f(u) = 2, or at least two vertices with The weight of an R2DF is the sum The minimum weight of an R2DF is called the Roman {2}-domination number and is denoted by A double Roman dominating function (DRDF) on G is a function such that for every vertex if f(v) = 0, then v has at least two neighbors with or one neighbor w with f(w) = 3, and if f(v) = 1, then v must have at least one neighbor w with The weight of a DRDF is the value The minimum weight of a DRDF is called the double Roman domination number and is denoted by Given an graph G and a positive integer k, the R2DP (DRDP) problem is to check whether G has an R2DF (DRDF) of weight at most k. In this article, we first show that the R2DP problem is NP-complete for star convex bipartite graphs, comb convex bipartite graphs and bisplit graphs. We also show that the DRDP problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Next, we show that are obtained in linear time for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. Finally, we propose a -approximation algorithm for the minimum Roman {2}-domination problem and -approximation algorithm for the minimum double Roman domination problem, where Δ is the maximum degree of G.

For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

For a simple, undirected and connected graph \(G = (V, E)\), a total Roman dominating function (TRDF) \(f : V \rightarrow \lbrace 0, 1, 2 \rbrace \) has the property that, every vertex u with \(f(u) = 0\) is adjacent to at least one vertex v for which \(f(v) = 2\) and the subgraph induced by the set of vertices labeled one or two has no isolated vertices. A total double Roman dominating function (TDRDF) on G is a function \(f : V \rightarrow \lbrace 0, 1, 2, 3 \rbrace \) such that for every vertex \(v \in V\) if \(f(v) = 0\), then v has at least two neighbors x, y with \(f(x) = f(y) = 2\) or one neighbor w with \(f(w) = 3\), and if \(f(v) = 1\), then v must have at least one neighbor w with \(f(w) \ge 2\) and the subgraph induced by the set \(\{u_i : f(u_i) \ge 1\}\) has no isolated vertices. The weight of a T(D)RDF f is the sum \(f(V) = \sum _{v \in V}f(v)\). The minimum total (double) Roman domination problem (MT(D)RDP) is to find a T(D)RDF of minimum weight of the input graph. In this article, we show that MTRDP and MTDRDP are polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a \(2 (\ln (\varDelta - 0.5) + 1.5)\)-approximation algorithm (APX-AL) for the MTRDP and \(3 (\ln (\varDelta - 0.5) + 1.5)\)-APX-AL for the MTDRDP, where \(\varDelta \) is the maximum degree of G, and show that the same cannot have \((1 - \delta ) \ln |V|\) ratio APX-AL for any \(\delta > 0\) unless \(P = NP\). Finally, we show that MT(D)RDP is APX-hard for graphs with \( \varDelta =5\).KeywordsTotal Roman dominationTotal double Roman dominationAPX-complete

Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.

For an integer k ≥ 1 , let f be a function that assigns labels from the set { 0 , 1 , … , k + 1 } to the vertices of a simple graph G = ( V , E ) . The active neighborhood AN(v) of a vertex v ∈ V ( G ) with respect to f is the set of all neighbors of v that are assigned non-zero values under f. A [ k ] -Roman dominating function ( [ k ] -RDF) is a function f : V ( G ) → { 0 , 1 , 2 , … , k + 1 } such that for every vertex v ∈ V ( G ) with f(v) < k, we have ∑ u ∈ N [ v ] f ( u ) ≥ | A N ( v ) | + k . The weight of a [ k ] -RDF is the sum of its function values over the whole set of vertices, and the [ k ] -Roman domination number γ [ k R ] ( G ) is the minimum weight of a [ k ] -RDF on G. In this paper we determine various bounds on the [ k ] -Roman domination number. In particular, we show that for any integer k ≥ 2 every connected graph G of order n ≥ 3 , satisfies γ [ k R ] ( G ) ≤ ( 2 k + 1 ) 4 n , and we characterize the graphs G attaining this bound. Moreover, we show that if T is a nontrivial tree, then γ [ k R ] ( T ) ≥ k γ ( T ) + 1 for every integer k ≥ 2 and we characterize the trees attaining the lower bound. Finally, we prove the NP-completeness of the [ k ] -Roman domination problem in bipartite and chordal graphs.

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 &gt; 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.

For a graph [Formula: see text] and a function [Formula: see text], let [Formula: see text] ([Formula: see text]) be the set of vertices assigned the value [Formula: see text] by [Formula: see text]. A perfect Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] is adjacent to exactly one vertex [Formula: see text]. A perfect double Roman dominating function of a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that (1) every vertex [Formula: see text] is adjacent to exactly two vertices in [Formula: see text] and no vertex in [Formula: see text], or exactly one vertex in [Formula: see text] and no vertex in [Formula: see text]; (2) If [Formula: see text], then [Formula: see text] is adjacent to exactly one vertex in [Formula: see text] and no vertex in [Formula: see text]. In this paper, we prove that the decision problem associated to perfect Roman domination problem is NP-complete even when restricted to double star convex bipartite graphs, and the decision problem associated to perfect double Roman domination problem is NP-complete for comb convex bipartite graphs. We also prove that the decision problem associated to a similar Roman domination variant, namely minimum independent Roman domination problem is NP-complete even when restricted to double star convex bipartite graphs.

For a simple, undirected, connected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called a weakly connected Roman dominating function (WCRDF) of [Formula: see text] with weight [Formula: see text]. (C1). For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text] and (C2). The graph with vertex set [Formula: see text] and edge set [Formula: see text] or [Formula: see text] or both[Formula: see text] 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 [Formula: see text]-approximation algorithm for the MWCRDP and show that the same cannot have [Formula: see text] ratio approximation algorithm for any [Formula: see text] unless [Formula: see text]. Next, we show that MWCRDP is APX-hard for graphs with [Formula: see text]. 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.

Roman domination in regular graphs

In the k red (blue) domination problem for a bipartite graph \(G=(X,Y,E)\), we seek a subset \(D \subseteq X\) (respectively \(D \subseteq Y\)) of cardinality at most k that dominates vertices of Y (respectively X). The decision version of this problem is \(\textsc {NP}\)-complete for perfect elimination bipartite graphs but solvable in polynomial time for chordal bipartite graphs. We present a linear time algorithm to solve the minimum cardinality red domination problem for convex bipartite graphs. The algorithm presented is faster and simpler than that in the literature. Due to the asymmetry in convex bipartite graphs, the algorithm does not extend to k blue domination. We present a linear time algorithm to solve the minimum cardinality blue domination problem for convex bipartite graphs.KeywordsConvex bipartite graphRed dominating setBlue dominating set