Graphs with small balanced decomposition numbers

A balanced coloring of a graph \(G\) is an ordered pair \((R,B)\) of disjoint subsets \(R,B \subseteq V(G)\) with \(|R|=|B|\) . The balanced decomposition number  \(f(G)\) of a connected graph \(G\) is the minimum integer \(f\) such that for any balanced coloring \((R,B)\) of \(G\) there is a partition \(\mathcal{P}\) of \(V(G)\) such that \(S\) induces a connected subgraph with \(|S| \le f\) and \(|S \cap R| = |S \cap B|\) for \(S \in \mathcal{P}\) . This paper gives a short proof for the result by Fujita and Liu (2010) that a graph \(G\) of \(n\) vertices has \(f(G)=3\) if and only if \(G\) is \(\lfloor \frac{n}{2} \rfloor \) -connected but is not a complete graph.     

Lower bounds for independence numbers of some locally sparse graphs

An \(m\) -distinct-coloring is a proper vertex-coloring \(c\) of a graph \(G\) if for each vertex \(v\in V\) , any color appears in at most one of \(N_0(v)\) , \(N_1(v)\) , \(\ldots \) , and \(N_m(v)\) , where \(N_i(v)\) is the set of vertices at distance \(i\) from \(v\) . In this note, we show that if \(G\) is \(C_{2m+1}\) -free which is assigned an \((m+1)\) -distinct-coloring \(c\) , then \(\alpha (G)c(G)^{1/m}\ge \Omega \Big (\sum _{v} c(v)^{1/m}\Big )\) , where \(c(G)\) is the number of colors used in \(c\) and \(c(v)\) is the number of different colors appearing in \(N_1(v)\) . Moreover, we obtain that if \(G\) has \(N\) vertices and it contains neither \(C_{2m+1}\) nor \(C_{2m}\) , then \(\alpha (G)\ge \Omega \big ((N\log N)^{m/(m+1)}\big )\) . The algorithm in the proof for the first result is random, and that for the second is constructive.     

A greedy algorithm for the fault-tolerant connected dominating set in a general graph

Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless network is an effective way to save energy and alleviate broadcasting storm. Since nodes may fail due to an accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. A node set \(C\) is an \(m\) -fold connected dominating set ( \(m\) -fold CDS) of graph \(G\) if every node in \(V(G)\setminus C\) has at least \(m\) neighbors in \(C\) and the subgraph of \(G\) induced by \(C\) is connected. In this paper, we will present a greedy algorithm to compute an \(m\) -fold CDS in a general graph, which has size at most \(2+\ln (\Delta +m-2)\) times that of a minimum \(m\) -fold CDS, where \(\Delta \) is the maximum degree of the graph. This result improves on the previous best known performance ratio of \(2H(\Delta +m-1)\) for this problem, where \(H(\cdot )\) is the Harmonic number.     

Notes on L(1,1) and L(2,1) labelings for n-cube

Suppose \(d\) is a positive integer. An \(L(d,1)\) -labeling of a simple graph \(G=(V,E)\) is a function \(f:V\rightarrow \mathbb{N }=\{0,1,2,{\ldots }\}\) such that \(|f(u)-f(v)|\ge d\) if \(d_G(u,v)=1\) ; and \(|f(u)-f(v)|\ge 1\) if \(d_G(u,v)=2\) . The span of an \(L(d,1)\) -labeling \(f\) is the absolute difference between the maximum and minimum labels. The \(L(d,1)\) -labeling number, \(\lambda _d(G)\) , is the minimum of span over all \(L(d,1)\) -labelings of \(G\) . Whittlesey et al. proved that \(\lambda _2(Q_n)\le 2^k+2^{k-q+1}-2,\) where \(n\le 2^k-q\) and \(1\le q\le k+1\) . As a consequence, \(\lambda _2(Q_n)\le 2n\) for \(n\ge 3\) . In particular, \(\lambda _2(Q_{2^k-k-1})\le 2^k-1\) . In this paper, we provide an elementary proof of this bound. Also, we study the \(L(1,1)\) -labeling number of \(Q_n\) . A lower bound on \(\lambda _1(Q_n)\) are provided and \(\lambda _1(Q_{2^k-1})\) are determined.     

2-Distance paired-dominating number of graphs

Let \(G=(V,E)\) be a simple graph without isolated vertices. For a positive integer \(k\) , a subset \(D\) of \(V(G)\) is a \(k\) -distance paired-dominating set if each vertex in \(V\setminus {D}\) is within distance \(k\) of a vertex in \(D\) and the subgraph induced by \(D\) contains a perfect matching. In this paper, we give some upper bounds on the 2-distance paired-dominating number in terms of the minimum and maximum degree, girth, and order.     

2-Rainbow domination number of Cartesian products: C_{n}\square C_{3} and C_{n}\square C_{5}

A function \(f:V(G)\rightarrow \mathcal P (\{1,\ldots ,k\})\) is called a \(k\) -rainbow dominating function of \(G\) (for short \(kRDF\) of \(G)\) if \( \bigcup \nolimits _{u\in N(v)}f(u)=\{1,\ldots ,k\},\) for each vertex \( v\in V(G)\) with \(f(v)=\varnothing .\) By \(w(f)\) we mean \(\sum _{v\in V(G)}\left|f(v)\right|\) and we call it the weight of \(f\) in \(G.\) The minimum weight of a \( kRDF\) of \(G\) is called the \(k\) -rainbow domination number of \(G\) and it is denoted by \(\gamma _{rk}(G).\) We investigate the \(2\) -rainbow domination number of Cartesian products of cycles. We give the exact value of the \(2\) -rainbow domination number of \(C_{n}\square C_{3}\) and we give the estimation of this number with respect to \(C_{n}\square C_{5},\) \((n\ge 3).\) Additionally, for \(n=3,4,5,6,\) we show that \(\gamma _{r2}(C_{n}\square C_{5})=2n.\)      

Adjacent vertex distinguishing edge colorings of planar graphs with girth at least five

An adjacent vertex distinguishing edge coloring of a graph \(G\) is a proper edge coloring of \(G\) such that any pair of adjacent vertices admit different sets of colors. The minimum number of colors required for such a coloring of \(G\) is denoted by \(\chi ^{\prime }_{a}(G)\) . In this paper, we prove that if \(G\) is a planar graph with girth at least 5 and \(G\) is not a 5-cycle, then \(\chi ^{\prime }_{a}(G)\le \Delta +2\) , where \(\Delta \) is the maximum degree of \(G\) . This confirms partially a conjecture in Zhang et al. (Appl Math Lett 15:623–626, 2002).     

On metric dimension of permutation graphs

The metric dimension \(\dim (G)\) of a graph \(G\) is the minimum number of vertices such that every vertex of \(G\) is uniquely determined by its vector of distances to the set of chosen vertices. Let \(G_1\) and \(G_2\) be disjoint copies of a graph \(G\) , and let \(\sigma : V(G_1) \rightarrow V(G_2)\) be a permutation. Then, a permutation graph \(G_{\sigma }=(V, E)\) has the vertex set \(V=V(G_1) \cup V(G_2)\) and the edge set \(E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}\) . We show that \(2 \le \dim (G_{\sigma }) \le n-1\) for any connected graph \(G\) of order \(n\) at least \(3\) . We give examples showing that neither is there a function \(f\) such that \(\dim (G) for all pairs \((G,\sigma )\) , nor is there a function \(g\) such that \(g(\dim (G))>\dim (G_{\sigma })\) for all pairs \((G, \sigma )\) . Further, we characterize permutation graphs \(G_{\sigma }\) satisfying \(\dim (G_{\sigma })=n-1\) when \(G\) is a complete \(k\) -partite graph, a cycle, or a path on \(n\) vertices.     

F_{3}-domination problem of graphs

Given a graph \(G\) and a set \(S\subseteq V(G),\) a vertex \(v\) is said to be \(F_{3}\) -dominated by a vertex \(w\) in \(S\) if either \(v=w,\) or \(v\notin S\) and there exists a vertex \(u\) in \(V(G)-S\) such that \(P:wuv\) is a path in \(G\) . A set \(S\subseteq V(G)\) is an \(F_{3}\) -dominating set of \(G\) if every vertex \(v\) is \(F_{3}\) -dominated by a vertex \(w\) in \(S.\) The \(F_{3}\) -domination number of \(G\) , denoted by \(\gamma _{F_{3}}(G)\) , is the minimum cardinality of an \(F_{3}\) -dominating set of \(G\) . In this paper, we study the \(F_{3}\) -domination of Cartesian product of graphs, and give formulas to compute the \(F_{3}\) -domination number of \(P_{m}\times P_{n}\) and \(P_{m}\times C_{n}\) for special \(m,n.\)      

Every planar graph with cycles of length neither 4 nor 5 is (1,1,0)-colorable

Let \(d_1, d_2,\dots ,d_k\) be \(k\) non-negative integers. A graph \(G\) is \((d_1,d_2,\ldots ,d_k)\) -colorable, if the vertex set of \(G\) can be partitioned into subsets \(V_1,V_2,\ldots ,V_k\) such that the subgraph \(G[V_i]\) induced by \(V_i\) has maximum degree at most \(d_i\) for \(i=1,2,\ldots ,k.\) Let \(\digamma \) be the family of planar graphs with cycles of length neither 4 nor 5. Steinberg conjectured that every graph of \(\digamma \) is \((0,0,0)\) -colorable. In this paper, we prove that every graph of \(\digamma \) is \((1,1,0)\) -colorable.     

Acyclic edge coloring of planar graphs without a 3-cycle adjacent to a 6-cycle

An acyclic edge coloring of a graph \(G\) is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index \(a'(G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an acyclic edge coloring using \(k\) colors. Fiam? ik (Math Slovaca 28:139–145, 1978) and later Alon et al. (J Graph Theory 37:157–167, 2001) conjectured that \(a'(G)\le \Delta +2\) for any simple graph \(G\) with maximum degree \(\Delta \) . In this paper, we confirm this conjecture for planar graphs without a \(3\) -cycle adjacent to a \(6\) -cycle.     

Minimum number of disjoint linear forests covering a planar graph

Graph coloring has interesting real-life applications in optimization, computer science and network design, such as file transferring in a computer network, computation of Hessians matrix and so on. In this paper, we consider one important coloring, linear arboricity, which is an improper edge coloring. Moreover, we study linear arboricity on planar graphs with maximum degree \(\varDelta \ge 7\) . We have proved that the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \) , if for each vertex \(v\in V(G)\) , there are two integers \(i_v,j_v\in \{3,4,5,6,7,8\}\) such that any two cycles of length \(i_v\) and \(j_v\) , which contain \(v\) , are not adjacent. Clearly, if \(i_v=i, j_v=j\) for each vertex \(v\in V(G)\) , then we can easily get one corollary: for two fixed integers \(i,j\in \{3,4,5,6,7,8\}\) , if there is no adjacent cycles with length \(i\) and \(j\) in \(G\) , then the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \) .     

Mobile facility location: combinatorial filtering via weighted occupancy

An instance of the mobile facility location problem consists of a complete directed graph \(G = (V, E)\) , in which each arc \((u, v) \in E\) is associated with a numerical attribute \(\mathcal M (u,v)\) , representing the cost of moving any object from \(u\) to \(v\) . An additional ingredient of the input is a collection of servers \(S = \{ s_1, \ldots , s_k \}\) and a set of clients \(C = \{ c_1, \ldots , c_\ell \}\) , which are located at nodes of the underlying graph. With this setting in mind, a movement scheme is a function \(\psi : S \rightarrow V\) that relocates each server \(s_i\) to a new position, \(\psi ( s_i )\) . We refer to \(\mathcal M ( s_i, \psi ( s_i ) )\) as the relocation cost of \(s_i\) , and to \(\min _{i \in [k]} \mathcal M (c_j, \psi ( s_i ) )\) , the cost of assigning client \(c_j\) to the nearest final server location, as the service cost of \(c_j\) . The objective is to compute a movement scheme that minimizes the sum of relocation and service costs. In this paper, we resolve an open question posed by Demaine et al. (SODA ’07) by characterizing the approximability of mobile facility location through LP-based methods. We also develop a more efficient algorithm, which is based on a combinatorial filtering approach. The latter technique is of independent interest, as it may be applicable in other settings as well. In this context, we introduce a weighted version of the occupancy problem, for which we establish interesting tail bounds, not before demonstrating that existing bounds cannot be extended.     

Efficient polynomial-time algorithms for the constrained LCS problem with strings exclusion

In this paper, we revisit a recent variant of the longest common subsequence (LCS) problem, the string-excluding constrained LCS (STR-EC-LCS) problem, which was first addressed by Chen and Chao (J Comb Optim 21(3):383–392, 2011). Given two sequences \(X\) and \(Y\) of lengths \(m\) and \(n,\) respectively, and a constraint string \(P\) of length \(r,\) we are to find a common subsequence \(Z\) of \(X\) and \(Y\) which excludes \(P\) as a substring and the length of \(Z\) is maximized. In fact, this problem cannot be correctly solved by the previously proposed algorithm. Thus, we give a correct algorithm with \(O(mnr)\) time to solve it. Then, we revisit the STR-EC-LCS problem with multiple constraints \(\{ P_1, P_2, \ldots , P_k \}.\) We propose a polynomial-time algorithm which runs in \(O(mnR)\) time, where \(R = \sum _{i=1}^{k} |P_i|,\) and thus it overthrows the previous claim of NP-hardness.     

Realizations of the game domination number

Domination game is a game on a finite graph which includes two players. First player, Dominator, tries to dominate a graph in as few moves as possible; meanwhile the second player, Staller, tries to hold him back and delay the end of the game as long as she can. In each move at least one additional vertex has to be dominated. The number of all moves in the game in which Dominator makes the first move and both players play optimally is called the game domination number and is denoted by \(\gamma _g\) . The total number of moves in a Staller-start game is denoted by \(\gamma _g^{\prime }\) . It is known that \(|\gamma _g(G)-\gamma _g^{\prime }(G)|\le 1\) for any graph \(G\) . Graph \(G\) realizes a pair \((k,l)\) if \(\gamma _g(G)=k\) and \(\gamma _g^{\prime }(G)=l\) . It is shown that pairs \((2k,2k-1)\) for all \(k\ge 2\) can be realized by a family of 2-connected graphs. We also present 2-connected classes which realize pairs \((k,k)\) and \((k,k+1)\) . Exact game domination number for combs and 1-connected realization of the pair \((2k+1,2k)\) are also given.     

On colour-blind distinguishing colour pallets in regular graphs

Consider a graph \(G=(V,E)\) and a colouring of its edges with \(k\) colours. Then every vertex \(v\in V\) is associated with a ‘pallet’ of incident colours together with their frequencies, which sum up to the degree of \(v\) . We say that two vertices have distinct pallets if they differ in frequency of at least one colour. This is always the case if these vertices have distinct degrees. We consider an apparently the worse case, when \(G\) is regular. Suppose further that this coloured graph is being examined by a person who cannot name any given colour, but distinguishes one from another. Could we colour the edges of \(G\) so that a person suffering from such colour-blindness is certain that colour pallets of every two adjacent vertices are distinct? Using the Lopsided Lovász Local Lemma, we prove that it is possible using 15 colours for every \(d\) -regular graph with \(d\ge 960\) .     

Efficient algorithms for the max k -vertex cover problem

Given a graph  \(G(V,E)\) of order  \(n\) and a constant \(k \leqslant n\) , the max  \(k\) -vertex cover problem consists of determining  \(k\) vertices that cover the maximum number of edges in  \(G\) . In its (standard) parameterized version, max  \(k\) -vertex cover can be stated as follows: “given  \(G,\) \(k\) and parameter  \(\ell ,\) does  \(G\) contain  \(k\) vertices that cover at least  \(\ell \) edges?”. We first devise moderately exponential exact algorithms for max  \(k\) -vertex cover, with time-complexity exponential in  \(n\) but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max  \(k\) -vertex cover with complexity bounded above by the maximum among  \(c^k\) and  \(\gamma ^{\tau },\) for some \(\gamma < 2,\) where  \(\tau \) is the cardinality of a minimum vertex cover of  \(G\) (note that \({\textsc {max}}\,\) k \({\textsc {\!-vertex cover}}{} \notin \mathbf{FPT}\) with respect to parameter  \(k\) unless \(\mathbf{FPT} = \mathbf{W[1]}\) ), using polynomial space. We finally study approximation of max  \(k\) -vertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch-up on polynomial inapproximability, by providing algorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time.     

Heterochromatic tree partition number in complete multipartite graphs

The heterochromatic tree partition number of an \(r\) -edge-colored graph \(G,\) denoted by \(t_r(G),\) is the minimum positive integer \(p\) such that whenever the edges of the graph \(G\) are colored with \(r\) colors, the vertices of \(G\) can be covered by at most \(p\) vertex disjoint heterochromatic trees. In this article we determine the upper and lower bounds for the heterochromatic tree partition number \(t_r(K_{n_1,n_2,\ldots ,n_k})\) of an \(r\) -edge-colored complete \(k\) -partite graph \(K_{n_1,n_2,\ldots ,n_k}\) , and the gap between upper and lower bounds is at most one.     

A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

Let \((MQP)\) be a general mixed-integer quadratic program that consists of minimizing a quadratic function \(f(x) = x^TQx +c^Tx\) subject to linear constraints. Our approach to solve \((MQP)\) is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables \(y_{ij}\) , additional quadratic constraints \(y_{ij}=x_ix_j\) , a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in Billionnet et al. (Math Program 131(1):381–401, 2012), the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints \(y_{ij}=x_ix_j\) to solve \((MQP)\) . Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with the approach of Billionnet et al. (2012) and with the general mixed-integer nonlinear solver BARON (Sahinidis and Tawarmalani, Global optimization of mixed-integer nonlinear programs, user’s manual, 2010).     

Recognition of overlap graphs

Overlap graphs occur in computational biology and computer science, and have applications in genome sequencing, string compression, and machine scheduling. Given two strings \(s_{i}\) and \(s_{j}\) , their overlap string is defined as the longest string \(v\) such that \(s_{i} = uv\) and \(s_{j} = vw\) , for some non empty strings \(u,w\) , and its length is called the overlap between these two strings. A weighted directed graph is an overlap graph if there exists a set of strings with one-to-one correspondence to the vertices of the graph, such that each arc weight in the graph equals the overlap between the corresponding strings. In this paper, we characterize the class of overlap graphs, and we present a polynomial time recognition algorithm as a direct consequence. Given a weighted directed graph \(G\) , the algorithm constructs a set of strings that has \(G\) as its overlap graph, or decides that this is not possible.