Disjunctive total domination in graphs

Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, \(\gamma _t(G)\). A set \(S\) of vertices in \(G\) is a disjunctive total dominating set of \(G\) if every vertex is adjacent to a vertex of \(S\) or has at least two vertices in \(S\) at distance \(2\) from it. The disjunctive total domination number, \(\gamma ^d_t(G)\), is the minimum cardinality of such a set. We observe that \(\gamma ^d_t(G) \le \gamma _t(G)\). We prove that if \(G\) is a connected graph of order \(n \ge 8\), then \(\gamma ^d_t(G) \le 2(n-1)/3\) and we characterize the extremal graphs. It is known that if \(G\) is a connected claw-free graph of order \(n\), then \(\gamma _t(G) \le 2n/3\) and this upper bound is tight for arbitrarily large \(n\). We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if \(G\) is a connected claw-free graph of order \(n > 14\), then \(\gamma ^d_t(G) \le 4n/7\) and we characterize the graphs achieving equality in this bound.     

Equality in a bound that relates the size and the restrained domination number of a graph

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a restrained dominating set if every vertex in \(V-S\) is adjacent to a vertex in \(S\) and to a vertex in \(V-S\). The restrained domination number of \(G\), denoted \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). The best possible upper bound \(q(n,k)\) is established in Joubert (Discrete Appl Math 161:829–837, 2013) on the size \(m(G)\) of a graph \(G\) with a given order \(n \ge 5\) and restrained domination number \(k \in \{3, \ldots , n-2\}\). We extend this result to include the cases \(k=1,2,n\), and characterize graphs \(G\) of order \(n \ge 1\) and restrained domination number \(k \in \{1,\dots , n-2,n\}\) for which \(m(G)=q(n,k)\).     

2-Edge connected dominating sets and 2-Connected dominating sets of a graph

A \(k\)-connected (resp. \(k\)-edge connected) dominating set \(D\) of a connected graph \(G\) is a subset of \(V(G)\) such that \(G[D]\) is \(k\)-connected (resp. \(k\)-edge connected) and each \(v\in V(G)\backslash D\) has at least one neighbor in \(D\). The \(k\) -connected domination number (resp. \(k\) -edge connected domination number) of a graph \(G\) is the minimum size of a \(k\)-connected (resp. \(k\)-edge connected) dominating set of \(G\), and denoted by \(\gamma _k(G)\) (resp. \(\gamma '_k(G)\)). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph \(G\), if it is \(2\)-edge connected, then \(\gamma '_2(G)\le 4\alpha (G)-1\), and it is \(2\)-connected, then \(\gamma _2(G)\le 6\alpha (G)-3\), where \(\alpha (G)\) is the independent number of \(G\).     

An inequality that relates the size of a bipartite graph with its order and restrained domination number

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a restrained dominating set if every vertex in \(V-S\) is adjacent to a vertex in \(S\) and to a vertex in \(V-S\). The restrained domination number of \(G\), denoted \(\gamma _{r}(G)\), is the smallest cardinality of a restrained dominating set of \(G\). Consider a bipartite graph \(G\) of order \(n\ge 4,\) and let \(k\in \{2,3,...,n-2\}.\) In this paper we will show that if \(\gamma _{r}(G)=k\), then \(m\le ((n-k)(n-k+6)+4k-8)/4\). We will also show that this bound is best possible.     

Signed total Roman domination in graphs

Let \(G\) be a finite and simple graph with vertex set \(V(G)\). A signed total Roman dominating function (STRDF) on a graph \(G\) is a function \(f:V(G)\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{x\in N(v)}f(x)\ge 1\) for each vertex \(v\in V(G)\), where \(N(v)\) is the neighborhood of \(v\), and (ii) every vertex \(u\) for which \(f(u)=-1\) is adjacent to at least one vertex \(v\) for which \(f(v)=2\). The weight of an SRTDF \(f\) is \(\sum _{v\in V(G)}f(v)\). The signed total Roman domination number \(\gamma _{stR}(G)\) of \(G\) is the minimum weight of an STRDF on \(G\). In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on \(\gamma _{stR}(G)\). In addition, we determine the signed total Roman domination number of some classes of graphs.     

Signed mixed Roman domination numbers in graphs

Let \(G = (V;E)\) be a simple graph with vertex set \(V\) and edge set \(E\). A signed mixed Roman dominating function (SMRDF) of \(G\) is a function \(f: V\cup E\rightarrow \{-1,1,2\}\) satisfying the conditions that (i) \(\sum _{y\in N_m[x]}f(y)\ge 1\) for each \(x\in V\cup E\), where \(N_m[x]\) is the set, called mixed closed neighborhood of \(x\), consists of \(x\) and the elements of \(V\cup E\) adjacent or incident to \(x\) (ii) every element \(x\in V\cup E\) for which \(f(x) = -1\) is adjacent or incident to at least one element \(y\in V\cup E\) for which \(f(y) = 2\). The weight of a SMRDF \(f\) is \(\omega (f)=\sum _{x\in V\cup E}f(x)\). The signed mixed Roman domination number \(\gamma _{sR}^*(G)\) of \(G\) is the minimum weight of a SMRDF of \(G\). In this paper we initiate the study of the signed mixed Roman domination number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.     

Characterisation of forests with trivial game domination numbers

In the domination game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set \(A_*\) dominates the whole graph. The Dominator plays to minimise the size of the set \(A_*\) while the Staller plays to maximise it. A graph is \(D\)-trivial if when the Dominator plays first and both players play optimally, the set \(A_*\) is a minimum dominating set of the graph. A graph is \(S\)-trivial if the same is true when the Staller plays first. We consider the problem of characterising \(D\)-trivial and \(S\)-trivial graphs. We give complete characterisations of \(D\)-trivial forests and of \(S\)-trivial forests. We also show that \(2\)-connected \(D\)-trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.     

An efficient algorithm for distance total domination in block graphs

The \(k\)-distance total domination problem is to find a minimum vertex set \(D\) of a graph such that every vertex of the graph is within distance \(k\) from some vertex of \(D\) other than itself, where \(k\) is a fixed positive integer. In the present paper, by using a labeling method, we design an efficient algorithm for solving the \(k\)-distance total domination problem on block graphs, a superclass of trees.     

Weak {2}-domination number of Cartesian products of cycles

For a graph \(G=(V, E)\), a weak \(\{2\}\)-dominating function \(f:V\rightarrow \{0,1,2\}\) has the property that \(\sum _{u\in N(v)}f(u)\ge 2\) for every vertex \(v\in V\) with \(f(v)= 0\), where N(v) is the set of neighbors of v in G. The weight of a weak \(\{2\}\)-dominating function f is the sum \(\sum _{v\in V}f(v)\) and the minimum weight of a weak \(\{2\}\)-dominating function is the weak \(\{2\}\)-domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak \(\{2\}\)-domination number of \(C_n \Box C_5\), which was obtained by St?pień, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak \(\{2\}\)-domination numbers of \(C_n \Box C_3\) and \(C_n \Box C_4\).     

Lambda number for the direct product of some family of graphs

An L(2, 1)-labeling for a graph \(G=(V,E)\) is a function f on V such that \(|f(u)-f(v)|\ge 2\) if u and v are adjacent and f(u) and f(v) are distinct if u and v are vertices of distance two. The L(2, 1)-labeling number, or the lambda number \(\lambda (G)\), for G is the minimum span over all L(2, 1)-labelings of G. When \(P_{m}\times C_{n}\) is the direct product of a path \(P_m\) and a cycle \(C_n\), Jha et al. (Discret Appl Math 145:317–325, 2005) computed the lambda number of \(P_{m}\times C_{n}\) for \(n\ge 3\) and \(m=4,5\). They also showed that when \(m\ge 6\) and \(n\ge 7\), \(\lambda (P_{m}\times C_{n})=6\) if and only if n is the multiple of 7 and conjectured that it is 7 if otherwise. They also showed that \(\lambda (C_{7i}\times C_{7j})=6\) for some i, j. In this paper, we show that when \(m\ge 6\) and \(n\ge 3\), \(\lambda (P_m\times C_n)=7\) if and only if n is not a multiple of 7. Consequently the conjecture is proved. Here we also provide the conditions on m and n such that \(\lambda (C_m\times C_n)\le 7\).     

Power domination with bounded time constraints

Based on the power observation rules, the problem of monitoring a power utility network can be transformed into the graph-theoretic power domination problem, which is an extension of the well-known domination problem. A set \(S\) is a power dominating set (PDS) of a graph \(G=(V,E)\) if every vertex \(v\) in \(V\) can be observed under the following two observation rules: (1) \(v\) is dominated by \(S\), i.e., \(v \in S\) or \(v\) has a neighbor in \(S\); and (2) one of \(v\)’s neighbors, say \(u\), and all of \(u\)’s neighbors, except \(v\), can be observed. The power domination problem involves finding a PDS with the minimum cardinality in a graph. Similar to message passing protocols, a PDS can be considered as a dominating set with propagation that applies the second rule iteratively. This study investigates a generalized power domination problem, which limits the number of propagation iterations to a given positive integer; that is, the second rule is applied synchronously with a bounded time constraint. To solve the problem in block graphs, we propose a linear time algorithm that uses a labeling approach. In addition, based on the concept of time constraints, we provide the first nontrivial lower bound for the power domination problem.     

Algorithmic aspects of <Emphasis Type="Italic">b</Emphasis>-disjunctive domination in graphs

For a fixed integer \(b>1\), a set \(D\subseteq V\) is called a b-disjunctive dominating set of the graph \(G=(V,E)\) if for every vertex \(v\in V{\setminus }D\), v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The Minimum b-Disjunctive Domination Problem (MbDDP) is to find a b-disjunctive dominating set of minimum cardinality. The cardinality of a minimum b-disjunctive dominating set of G is called the b-disjunctive domination number of G, and is denoted by \(\gamma _{b}^{d}(G)\). Given a positive integer k and a graph G, the b-Disjunctive Domination Decision Problem (bDDDP) is to decide whether G has a b-disjunctive dominating set of cardinality at most k. In this paper, we first show that for a proper interval graph G, \(\gamma _{b}^{d}(G)\) is equal to \(\gamma (G)\), the domination number of G for \(b \ge 3\) and observe that \(\gamma _{b}^{d}(G)\) need not be equal to \(\gamma (G)\) for \(b=2\). We then propose a polynomial time algorithm to compute a minimum cardinality b-disjunctive dominating set of a proper interval graph for \(b=2\). Next we tighten the NP-completeness of bDDDP by showing that it remains NP-complete even in chordal graphs. We also propose a \((\ln ({\varDelta }^{2}+(b-1){\varDelta }+b)+1)\)-approximation algorithm for MbDDP, where \({\varDelta }\) is the maximum degree of input graph \(G=(V,E)\) and prove that MbDDP cannot be approximated within \((1-\epsilon ) \ln (|V|)\) for any \(\epsilon >0\) unless NP \(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we show that MbDDP is APX-complete for bipartite graphs with maximum degree \(\max \{b,4\}\).     

On the efficiency index of a graph

A graph \(G\) has an efficient dominating set \(D \subseteq V(G)\) if \(D\) dominates every vertex exactly once. In this paper we introduce the study of the family \({S_k}\) of graphs for which every \(G-S\) is efficiently dominatable for \(0 \le |S|\le k\). Assuming that \(G\) is efficiently dominatable, the efficiency index is the largest value k for which \(G\) is in \(S_k\). A graph \(G\) will be called super-efficient if every induced subgraph is efficiently dominatable. We give some characterizations for trees, grids, cylinders and torii to be super-efficient.     

A greedy algorithm for the minimum 2-connected m-fold dominating set problem

To save energy and alleviate interference in a wireless sensor network, connected dominating set (CDS) has been proposed as the virtual backbone. Since nodes may fail due to accidental damage or energy depletion, it is desirable to construct a fault tolerant CDS, which can be modeled as a \(k\)-connected \(m\)-fold dominating set \(((k,m)\)-CDS for short): a subset of nodes \(C\subseteq V(G)\) is a \((k,m)\)-CDS of \(G\) if every node in \(V(G)\setminus C\) is adjacent with at least \(m\) nodes in \(C\) and the subgraph of \(G\) induced by \(C\) is \(k\)-connected.In this paper, we present an approximation algorithm for the minimum \((2,m)\)-CDS problem with \(m\ge 2\). Based on a \((1,m)\)-CDS, the algorithm greedily merges blocks until the connectivity is raised to two. The most difficult problem in the analysis is that the potential function used in the greedy algorithm is not submodular. By proving that an optimal solution has a specific decomposition, we managed to prove that the approximation ratio is \(\alpha +2(1+\ln \alpha )\), where \(\alpha \) is the approximation ratio for the minimum \((1,m)\)-CDS problem. This improves on previous approximation ratios for the minimum \((2,m)\)-CDS problem, both in general graphs and in unit disk graphs.     

Neighbor sum distinguishing total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

A total coloring of a graph \(G\) is a coloring of its vertices and edges such that adjacent or incident vertices and edges are not colored with the same color. A total \([k]\)-coloring of a graph \(G\) is a total coloring of \(G\) by using the color set \([k]=\{1,2,\ldots ,k\}\). Let \(f(v)\) denote the sum of the colors of a vertex \(v\) and the colors of all incident edges of \(v\). A total \([k]\)-neighbor sum distinguishing-coloring of \(G\) is a total \([k]\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(f(u)\ne f(v)\). Let \(G\) be a graph which can be embedded in a surface of nonnegative Euler characteristic. In this paper, it is proved that the total neighbor sum distinguishing chromatic number of \(G\) is \(\Delta (G)+2\) if \(\Delta (G)\ge 14\), where \(\Delta (G)\) is the maximum degree of \(G\).     

Special Issue for FAW 2014

For a graph \(G=(V,E)\), a dominating set is a set \(D\subseteq V\) such that every vertex \(v\in V\setminus D\) has a neighbor in \(D\). The minimum outer-connected dominating set (Min-Outer-Connected-Dom-Set) problem for a graph \(G\) is to find a dominating set \(D\) of \(G\) such that \(G[V\setminus D]\), the induced subgraph by \(G\) on \(V\setminus D\), is connected and the cardinality of \(D\) is minimized. In this paper, we consider the complexity of the Min-Outer-Connected-Dom-Set problem. In particular, we show that the decision version of the Min-Outer-Connected-Dom-Set problem is NP-complete for split graphs, a well known subclass of chordal graphs. We also consider the approximability of the Min-Outer-Connected-Dom-Set problem. We show that the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of \((1-\varepsilon ) \ln |V|\) for any \(\varepsilon >0\), unless NP \(\subseteq \) DTIME(\(|V|^{\log \log |V|}\)). For sufficiently large values of \(\varDelta \), we show that for graphs with maximum degree \(\varDelta \), the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of \(\ln \varDelta -C \ln \ln \varDelta \) for some constant \(C\), unless P \(=\) NP. On the positive side, we present a \(\ln (\varDelta +1)+1\)-factor approximation algorithm for the Min-Outer-Connected-Dom-Set problem for general graphs. We show that the Min-Outer-Connected-Dom-Set problem is APX-complete for graphs of maximum degree 4.     

$$$$-labeling for brick product graphs

Let \(G=(V, E)\) be a graph. Denote \(d_G(u, v)\) the distance between two vertices \(u\) and \(v\) in \(G\). An \(L(2, 1)\)-labeling of \(G\) is a function \(f: V \rightarrow \{0,1,\cdots \}\) such that for any two vertices \(u\) and \(v\), \(|f(u)-f(v)| \ge 2\) if \(d_G(u, v) = 1\) and \(|f(u)-f(v)| \ge 1\) if \(d_G(u, v) = 2\). The span of \(f\) is the difference between the largest and the smallest number in \(f(V)\). The \(\lambda \)-number of \(G\), denoted \(\lambda (G)\), is the minimum span over all \(L(2,1 )\)-labelings of \(G\). In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716–736, 2013) in the case when (i) \(\ell \) is even, or (ii) \(\ell \ge 5\) is odd and \(0 \le r \le 8\).     

$$$$-labeling number of Cartesian product of path and cycle

For positive numbers \(j\) and \(k\), an \(L(j,k)\)-labeling \(f\) of \(G\) is an assignment of numbers to vertices of \(G\) such that \(|f(u)-f(v)|\ge j\) if \(d(u,v)=1\), and \(|f(u)-f(v)|\ge k\) if \(d(u,v)=2\). The span of \(f\) is the difference between the maximum and the minimum numbers assigned by \(f\). The \(L(j,k)\)-labeling number of \(G\), denoted by \(\lambda _{j,k}(G)\), is the minimum span over all \(L(j,k)\)-labelings of \(G\). In this article, we completely determine the \(L(j,k)\)-labeling number (\(2j\le k\)) of the Cartesian product of path and cycle.     

A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Let \(G=(V,E)\) be a graph and \(\phi \) be a total \(k\)-coloring of \(G\) using the color set \(\{1,\ldots , k\}\). Let \(\sum _\phi (u)\) denote the sum of the color of the vertex \(u\) and the colors of all incident edges of \(u\). A \(k\)-neighbor sum distinguishing total coloring of \(G\) is a total \(k\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(\sum _\phi (u)\ne \sum _\phi (v)\). By \(\chi ^{''}_{nsd}(G)\), we denote the smallest value \(k\) in such a coloring of \(G\). Pil?niak and Wo?niak first introduced this coloring and conjectured that \(\chi _{nsd}^{''}(G)\le \Delta (G)+3\) for any simple graph \(G\). In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with \(\Delta (G)\ge 7\). Moreover, we also show that \(\chi _{nsd}^{''}(G)\le \Delta (G)+2\) for planar graphs without intersecting triangles with \(\Delta (G) \ge 9\). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.