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.     

The 3-rainbow index and connected dominating sets

A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of \(G\) is called a 3-rainbow coloring if for any three vertices in \(G\), there exists a rainbow tree connecting them. The 3-rainbow index \(rx_3(G)\) of \(G\) is defined as the minimum number of colors that are needed in a 3-rainbow coloring of \(G\). This concept, introduced by Chartrand et al., can be viewed as a generalization of the rainbow connection. In this paper, we study the 3-rainbow index by using connected 3-way dominating sets and 3-dominating sets. We show that for every connected graph \(G\) on \(n\) vertices with minimum degree at least \(\delta \, (3\le \delta \le 5)\), \(rx_{3}(G)\le \frac{3n}{\delta +1}+4\), and the bound is tight up to an additive constant; whereas for every connected graph \(G\) on \(n\) vertices with minimum degree at least \(\delta \, (\delta \ge 3)\), we get that \(rx_{3}(G)\le \frac{\ln (\delta +1)}{\delta +1}(1+o_{\delta }(1))n+5\). In addition, we obtain some tight upper bounds of the 3-rainbow index for some special graph classes, including threshold graphs, chain graphs and interval graphs.     

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.     

On minimally 2-connected graphs with generalized connectivity $$\kappa _{3}=2$$

For \(S\subseteq G\), let \(\kappa (S)\) denote the maximum number r of edge-disjoint trees \(T_1, T_2, \ldots , T_r\) in G such that \(V(T_i)\cap V(T_j)=S\) for any \(i,j\in \{1,2,\ldots ,r\}\) and \(i\ne j\). For every \(2\le k\le n\), the k-connectivity of G, denoted by \(\kappa _k(G)\), is defined as \(\kappa _k(G)=\hbox {min}\{\kappa (S)| S\subseteq V(G)\ and\ |S|=k\}\). Clearly, \(\kappa _2(G)\) corresponds to the traditional connectivity of G. In this paper, we focus on the structure of minimally 2-connected graphs with \(\kappa _{3}=2\). Denote by \(\mathcal {H}\) the set of minimally 2-connected graphs with \(\kappa _{3}=2\). Let \(\mathcal {B}\subseteq \mathcal {H}\) and every graph in \(\mathcal {B}\) is either \(K_{2,3}\) or the graph obtained by subdividing each edge of a triangle-free 3-connected graph. We obtain that \(H\in \mathcal {H}\) if and only if \(H\in \mathcal {B}\) or H can be constructed from one or some graphs \(H_{1},\ldots ,H_{k}\) in \(\mathcal {B}\) (\(k\ge 1\)) by applying some operations recursively.     

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.     

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 \(G\).     

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.     

The 2-surviving rate of planar graphs without 5-cycles

Let \(G\) be a connected graph with \(n\ge 2\) vertices. Let \(k\ge 1\) be an integer. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects \(k\)-vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let \(\hbox {sn}_k(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The \(k\)-surviving rate \(\rho _k(G)\) of \(G\) is defined to be \(\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)\), which is the average proportion of saved vertices. In this paper, we prove that if \(G\) is a planar graph with \(n\ge 2\) vertices and without 5-cycles, then \(\rho _2(G)>\frac{1}{363}\).     

Reconfiguration of dominating sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of \(D_k(G)\), the graph consisting of a node for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that \(D_{\varGamma (G)+1}(G)\) is not necessarily connected, for \(\varGamma (G)\) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for \(b \ge 3\). Moreover, we construct an infinite family of graphs such that \(D_{\gamma (G)+1}(G)\) has exponential diameter, for \(\gamma (G)\) the minimum size of a dominating set. On the positive side, we show that \(D_{n-\mu }(G)\) is connected and of linear diameter for any graph G on n vertices with a matching of size at least \(\mu +1\).     

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 \(m(G)=q(n,k)\).     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).     

Neighbor sum distinguishing total coloring of graphs with bounded treewidth

A proper total k-coloring \(\phi \) of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\dots , k\}\) such that no adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let \(m_{\phi }(v)\) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if \(m_{\phi }(u)\not =m_{\phi }(v)\) for each edge \(uv\in E(G).\) Let \(\chi _{\Sigma }^t(G)\) be the neighbor sum distinguishing total chromatic number of a graph G. Pil?niak and Wo?niak conjectured that for any graph G, \(\chi _{\Sigma }^t(G)\le \Delta (G)+3\). In this paper, we show that if G is a graph with treewidth \(\ell \ge 3\) and \(\Delta (G)\ge 2\ell +3\), then \(\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1\). This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when \(\ell =3\) and \(\Delta \ge 9\), we show that \(\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2\) and characterize graphs with equalities.     

On the difference of two generalized connectivities of a graph

The concept of k-connectivity \(\kappa '_{k}(G)\) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity \(\kappa _{k}(G)\), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if \(\kappa '_k(G)\ne n-k+1\) where \(k\ge 3\), then \(0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1\); otherwise, \(-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k\). Moreover, all of these bounds are sharp. Some specific study is focused for the case \(k=3\). As results, we characterize the graphs with \(\kappa '_3(G)=\kappa _3(G)=t\) for \(t\in \{1, n-3, n-2\}\), and give a necessary condition for \(\kappa '_3(G)=\kappa _3(G)\) by showing that for a connected graph G of order n and size m, if \(\kappa '_3(G)=\kappa _3(G)=t\) where \(1\le t\le n-3\), then \(m\le {n-2\atopwithdelims ()2}+2t\). Moreover, the unique extremal graph is given for the equality to hold.     

Cacti with the smallest,second smallest,and third smallest Gutman index

The Gutman index (also known as Schultz index of the second kind) of a graph \(G\) is defined as \(Gut(G)=\sum \nolimits _{u,v\in V(G)}d(u)d(v)d(u, v)\). A graph \(G\) is called a cactus if each block of \(G\) is either an edge or a cycle. Denote by \(\mathcal {C}(n, k)\) the set of connected cacti possessing \(n\) vertices and \(k\) cycles. In this paper, we give the first three smallest Gutman indices among graphs in \(\mathcal {C}(n, k)\), the corresponding extremal graphs are characterized as well.     

An improved bound on 2-distance coloring plane graphs with girth 5

A vertex coloring is called \(2\)-distance if any two vertices at distance at most \(2\) from each other get different colors. The minimum number of colors in 2-distance colorings of \(G\) is its 2-distance chromatic number, denoted by \(\chi _2(G)\). Let \(G\) be a plane graph with girth at least \(5\). In this paper, we prove that \(\chi _2(G)\le \Delta +8\) for arbitrary \(\Delta (G)\), which partially improves some known results.     

Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph \(G=(V_G, E_G)\) is defined as \(\bar{R}(G)=\sum _{u,v \in V_G}(\delta _G(u)+\delta _G(v))\frac{1}{d_G(u,v)}\), where \(\delta _G(u)\) is the vertex degree of \(u\), and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). The reciprocal degree distance is an additive weight version of the Harary index, which is defined as \(H(G)=\sum _{u,v \in V_G}\frac{1}{d_G(u,v)}\). In this paper, the extremal \(\bar{R}\)-values on several types of important graphs are considered. The graph with the maximum \(\bar{R}\)-value among all the simple connected graphs of diameter \(d\) is determined. Among the connected bipartite graphs of order \(n\), the graph with a given matching number (resp. vertex connectivity) having the maximum \(\bar{R}\)-value is characterized. Finally, sharp upper bounds on \(\bar{R}\)-value among all simple connected outerplanar (resp. planar) graphs are determined.     

Neighbor sum distinguishing total coloring of 2-degenerate graphs

A proper k-total coloring of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\ldots ,k\}\) such that no two adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let f(v) denote the sum of the colors on the edges incident with v and the color on vertex v. A proper k-total coloring of G is called neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). Let \(\chi ''_{\Sigma }(G)\) denote the smallest integer k in such a coloring of G. Pil?niak and Wo?niak conjectured that for any graph G, \(\chi ''_{\Sigma }(G)\le \Delta (G)+3\). In this paper, we show that if G is a 2-degenerate graph, then \(\chi ''_{\Sigma }(G)\le \Delta (G)+3\); Moreover, if \(\Delta (G)\ge 5\) then \(\chi ''_{\Sigma }(G)\le \Delta (G)+2\).     

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.     

Some extremal properties of the multiplicatively weighted Harary index of a graph

Let \(G=(V_G, E_G)\) be a simple connected graph. The multiplicatively weighted Harary index of \(G\) is defined as \(H_M(G)=\sum _{\{u,v\}\subseteq V_G}\delta _G(u)\delta _G(v)\frac{1}{d_G(u,v)},\) where \(\delta _G(u)\) is the vertex degree of \(u\) and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G.\) This novel invariant is in fact the modification of the Harary index in which the contributions of vertex pairs are weighted by the product of their degrees. Deng et al. (J Comb Optim 2014, doi: 10.1007/s10878-013-9698-5) determined the extremal values on \(H_M\) of graphs among \(n\)-vertex trees (resp. unicyclic graphs). In this paper, as a continuance of it, the monotonicity of \(H_M(G)\) under some graph transformations were studied. Using these nice mathematical properties, the extremal graphs among \(n\)-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. are characterized, respectively. Some sharp upper bounds on the multiplicatively weighted Harary index of trees with given parameters are determined.     

Atoms of cyclic edge connectivity in regular graphs

A cyclic edge-cut of a connected graph \(G\) is an edge set, the removal of which separates two cycles. If \(G\) has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph \(G\), the cyclic edge connectivity of a graph \(G\), denoted by \(\lambda _c(G)\), is the minimum cardinality over all cyclic edge cuts. Let \(X\) be a non-empty proper subset of \(V(G)\). If \([X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}\) is a minimum cyclic edge cut of \(G\), then \(X\) is called a \(\lambda _c\) -fragment of \(G\). A \(\lambda _c\)-fragment with minimum cardinality is called a \(\lambda _c\) -atom. Let \(G\) be a \(k (k\ge 3)\)-regular cyclically separable graph with \(\lambda _c(G)<g(k-2)\), where \(g\) is the girth of \(G\). A combination of the results of Nedela and Skoviera (Math Slovaca 45:481–499, 1995) and Xu and Liu (Australas J Combin 30:41–49, 2004) gives that if \(k\ne 5\) then any two distinct \(\lambda _c\)-atoms of \(G\) are disjoint. The remaining case of \(k=5\) is considered in this paper, and a new proof for Nedela and ?koviera’s result is also given. As a result, we obtain the following result. If \(X\) and \(X'\) are two distinct \(\lambda _c\)-atoms of \(G\) such that \(X\cap X'\ne \emptyset \), then \((k,g)=(5,3)\) and \(G[X]\cong K_4\). As corollaries, several previous results are easily obtained.