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.     

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.     

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 \(G\).     

Improved approximations for buy-at-bul and shallow-light $$$$-Steiner trees and $$(,2)$$(,2)-subgraph

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light \(k\)-Steiner tree problem (SL\(k\)ST), we are given an undirected graph \(G=(V,E)\) with terminals \(T\subseteq V\) containing a root \(r\in T\), a cost function \(c:E\rightarrow \mathbb {R}^+\), a length function \(\ell :E\rightarrow \mathbb {R}^+\), a bound \(L>0\) and an integer \(k\ge 1\). The goal is to find a minimum \(c\)-cost \(r\)-rooted Steiner tree containing at least \(k\) terminals whose diameter under \(\ell \) metric is at most \(L\). The input to the buy-at-bulk \(k\)-Steiner tree problem (BB\(k\)ST) is similar: graph \(G=(V,E)\), terminals \(T\subseteq V\) containing a root \(r\in T\), cost and length functions \(c,\ell :E\rightarrow \mathbb {R}^+\), and an integer \(k\ge 1\). The goal is to find a minimum total cost \(r\)-rooted Steiner tree \(H\) containing at least \(k\) terminals, where the cost of each edge \(e\) is \(c(e)+\ell (e)\cdot f(e)\) where \(f(e)\) denotes the number of terminals whose path to root in \(H\) contains edge \(e\). We present a bicriteria \((O(\log ^2 n),O(\log n))\)-approximation for SL\(k\)ST: the algorithm finds a \(k\)-Steiner tree with cost at most \(O(\log ^2 n\cdot \text{ opt }^*)\) where \(\text{ opt }^*\) is the cost of an LP relaxation of the problem and diameter at most \(O(L\cdot \log n)\). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio \((O(\log ^4 n), O(\log ^2 n))\). Using this, we obtain an \(O(\log ^3 n)\)-approximation for BB\(k\)ST, which improves upon the \(O(\log ^4 n)\)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost \(2\)-edge-connected subgraph with at least \(k\) vertices, which is introduced as the \((k,2)\)-subgraph problem in Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the \(k\)-MST and the minimum cost \(2\)-edge-connected subgraph problems. We give an \(O(\log n)\)-approximation algorithm for this problem which improves upon the \(O(\log ^2 n)\)-approximation algorithm of Lau et al. (2009).     

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 \(m(G)=q(n,k)\).     

Total and paired domination numbers of $$C_m$$ bundles over a cycle $$C_n$$Cn

Let \(G=(V,E)\) be a simple graph without isolated vertices. A set \(S\) of vertices is a total dominating set of a graph \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\). A paired dominating set of \(G\) is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369–378, 2014) computed the exact values of total and paired domination numbers of Cartesian product \(C_n\square C_m\) for \(m=3,4\). Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369–378, 2014) to graph bundle and compute the total domination number and the paired domination number of \(C_m\) bundles over a cycle \(C_n\) for \(m=3,4\). Moreover, we give the exact value for the total domination number of Cartesian product \(C_n\square C_5\) and some upper bounds of \(C_m\) bundles over a cycle \(C_n\) where \(m\ge 5\).     

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.     

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 \(G\).     

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.     

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 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.     

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.     

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.     

Partial degree bounded edge pacing problem for graphs and $$$$-uniform hypergraphs

Given a graph \(G=(V,E)\) and a non-negative integer \(c_u\) for each \(u\in V\), partial degree bounded edge packing problem is to find a subgraph \(G^{\prime }=(V,E^{\prime })\) with maximum \(|E^{\prime }|\) such that for each edge \((u,v)\in E^{\prime }\), either \(deg_{G^{\prime }}(u)\le c_u\) or \(deg_{G^{\prime }}(v)\le c_v\). The problem has been shown to be NP-hard even for uniform degree constraint (i.e., all \(c_u\) being equal). In this work we study the general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors \(4\) and \(2\). Then we give a \(\log _2 n\) approximation algorithm for edge-weighted version of the problem and an efficient exact algorithm for edge-weighted trees with time complexity \(O(n\log n)\). We also consider a generalization of this problem to \(k\)-uniform hypergraphs and present a constant factor approximation algorithm based on linear programming using Lagrangian relaxation.     

Constrained pairwise and center-star sequences alignment problems

This paper considers the minimax regret vertex 2-sink location problem in a dynamic path network with positive edge lengths and uniform edge capacity. Let \(P\) be an undirected path graph of \(n\) vertices, and the weight (initial supply) of every vertex is known as an interval. The problem is to find two vertices \(x\) and \(y\) as two sinks on the path such that all the weights can evacuate to \(x\) and \(y\) with minimum regret of evacuation time in case of an emergency for any possible weight distribution. We present an \(O(n^3\log n)\) time algorithm.     

On (s,t)-relaxed L(2,1)-labeling of graphs

We initiate the study of relaxed \(L(2,1)\)-labelings of graphs. Suppose \(G\) is a graph. Let \(u\) be a vertex of \(G\). A vertex \(v\) is called an \(i\)-neighbor of \(u\) if \(d_G(u,v)=i\). A \(1\)-neighbor of \(u\) is simply called a neighbor of \(u\). Let \(s\) and \(t\) be two nonnegative integers. Suppose \(f\) is an assignment of nonnegative integers to the vertices of \(G\). If the following three conditions are satisfied, then \(f\) is called an \((s,t)\)-relaxed \(L(2,1)\)-labeling of \(G\): (1) for any two adjacent vertices \(u\) and \(v\) of \(G, f(u)\not =f(v)\); (2) for any vertex \(u\) of \(G\), there are at most \(s\) neighbors of \(u\) receiving labels from \(\{f(u)-1,f(u)+1\}\); (3) for any vertex \(u\) of \(G\), the number of \(2\)-neighbors of \(u\) assigned the label \(f(u)\) is at most \(t\). The minimum span of \((s,t)\)-relaxed \(L(2,1)\)-labelings of \(G\) is called the \((s,t)\)-relaxed \(L(2,1)\)-labeling number of \(G\), denoted by \(\lambda ^{s,t}_{2,1}(G)\). It is clear that \(\lambda ^{0,0}_{2,1}(G)\) is the so called \(L(2,1)\)-labeling number of \(G\). \(\lambda ^{1,0}_{2,1}(G)\) is simply written as \(\widetilde{\lambda }(G)\). This paper discusses basic properties of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of graphs. For any two nonnegative integers \(s\) and \(t\), the exact values of \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of paths, cycles and complete graphs are determined. Tight upper and lower bounds for \((s,t)\)-relaxed \(L(2,1)\)-labeling numbers of complete multipartite graphs and trees are given. The upper bounds for \((s,1)\)-relaxed \(L(2,1)\)-labeling number of general graphs are also investigated. We introduce a new graph parameter called the breaking path covering number of a graph. A breaking path \(P\) is a vertex sequence \(v_1,v_2,\ldots ,v_k\) in which each \(v_i\) is adjacent to at least one vertex of \(v_{i-1}\) and \(v_{i+1}\) for \(i=2,3,\ldots ,k-1\). A breaking path covering of \(G\) is a set of disjoint such vertex sequences that cover all vertices of \(G\). The breaking path covering number of \(G\), denoted by \(bpc(G)\), is the minimum number of breaking paths in a breaking path covering of \(G\). In this paper, it is proved that \(\widetilde{\lambda }(G)= n+bpc(G^{c})-2\) if \(bpc(G^{c})\ge 2\) and \(\widetilde{\lambda }(G)\le n-1\) if and only if \(bpc(G^{c})=1\). The breaking path covering number of a graph is proved to be computable in polynomial time. Thus, if a graph \(G\) is of diameter two, then \(\widetilde{\lambda }(G)\) can be determined in polynomial time. Several conjectures and problems on relaxed \(L(2,1)\)-labelings are also proposed.     

1.61-approximation for min-power strong connectivity with two power levels

Given a directed simple graph \(G=(V,E)\) and a cost function \(c:E \rightarrow R_+\), the power of a vertex \(u\) in a directed spanning subgraph \(H\) is given by \(p_H(u) = \max _{uv \in E(H)} c(uv)\), and corresponds to the energy consumption required for wireless node \(u\) to transmit to all nodes \(v\) with \(uv \in E(H)\). The power of \(H\) is given by \(p(H) = \sum _{u \in V} p_H(u)\). Power Assignment seeks to minimize \(p(H)\) while \(H\) satisfies some connectivity constraint. In this paper, we assume \(E\) is bidirected (for every directed edge \(e \in E\), the opposite edge exists and has the same cost), while \(H\) is required to be strongly connected. Moreover, we assume \(c:E \rightarrow \{A,B\}\), where \(0 \le A < B\). We improve the best known approximation ratio from 1.75 (Chen et al. IEEE GLOBECOM 2005) to \(\pi ^2/6 - 1/36 + \epsilon \le 1.61\) using an adaptation of the algorithm developed by Khuller et al. [SIAM J Comput 24(4):859–872 1995, Discr Appl Math 69(3):281–289 1996] for (unweighted) Minimum Strongly Connected Subgraph.     

An approximation algorithm for maximum weight budgeted connected set cover

This paper studies approximation algorithm for the maximum weight budgeted connected set cover (MWBCSC) problem. Given an element set \(X\), a collection of sets \({\mathcal {S}}\subseteq 2^X\), a weight function \(w\) on \(X\), a cost function \(c\) on \({\mathcal {S}}\), a connected graph \(G_{\mathcal {S}}\) (called communication graph) on vertex set \({\mathcal {S}}\), and a budget \(L\), the MWBCSC problem is to select a subcollection \({\mathcal {S'}}\subseteq {\mathcal {S}}\) such that the cost \(c({\mathcal {S'}})=\sum _{S\in {\mathcal {S'}}}c(S)\le L\), the subgraph of \(G_{\mathcal {S}}\) induced by \({\mathcal {S'}}\) is connected, and the total weight of elements covered by \({\mathcal {S'}}\) (that is \(\sum _{x\in \bigcup _{S\in {\mathcal {S'}}}S}w(x)\)) is maximized. We present a polynomial time algorithm for this problem with a natural communication graph that has performance ratio \(O((\delta +1)\log n)\), where \(\delta \) is the maximum degree of graph \(G_{\mathcal {S}}\) and \(n\) is the number of sets in \({\mathcal {S}}\). In particular, if every set has cost at most \(L/2\), the performance ratio can be improved to \(O(\log n)\).     

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.