The minimum vulnerability problem on specific graph classes

Suppose that each edge e of an undirected graph G is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of e, respectively. Then, we consider the problem of finding \(k\) paths in G between two prescribed vertices with the minimum total cost; each edge e can be shared without any cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost for e. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.     

An $$O^{*}(1.4366^n)$$-time exact algorithm for maximum $$P_2$$P2-packing in cubic graphs

Given a graph \(G=(V, E)\), a \(P_2\)-packing \(\mathcal {P}\) is a collection of vertex disjoint copies of \(P_2\)s in \(G\) where a \(P_2\) is a simple path with three vertices and two edges. The Maximum \(P_2\)-Packing problem is to find a \(P_2\)-packing \(\mathcal {P}\) in the input graph \(G\) of maximum cardinality. This problem is NP-hard for cubic graphs. In this paper, we give a branch-and-reduce algorithm for the Maximum \(P_2\)-Packing problem in cubic graphs. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time \(O^{*}(1.4366^n)\) which is faster than previous known exact algorithms where \(n\) is the number of vertices in the input graph.     

Finding paths with minimum shared edges

Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s to t in G so as to minimize the number of edges shared among the paths. This is a generalization of the well-known disjoint paths problem. While disjoint paths can be computed efficiently, we show that finding paths with minimum shared edges is NP-hard. Moreover, we show that it is even hard to approximate the minimum number of shared edges within a factor of $2^{\log^{1-\varepsilon}n}$ , for any constant ε>0. On the positive side, we show that there exists a (k?1)-approximation algorithm for the problem, using an adaption of a network flow algorithm. We design some heuristics to improve the quality of the output, and provide empirical results.     

On the algorithmic aspects of strong subcoloring

A partition of the vertex set V(G) of a graph G into \(V(G)=V_1\cup V_2\cup \cdots \cup V_k\) is called a k-strong subcoloring if \(d(x,y)\ne 2\) in G for every \(x,y\in V_i\), \(1\le i \le k\) where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as \(\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k\)-\(\text {strong subcoloring}\}\). In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of \(K_{1,r}\)-free split graphs, StrongSubcoloring is in P when \(r\le 3\) and NP-complete when \(r>3\) and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of \(P_4\); otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every \(x,y\in S\), \(d(x,y)= 2\) in G and the cardinality of a maximum strong set in G is denoted by \(\alpha _{s}(G)\). Clearly, \(\alpha _{s}(G)\le \chi _{sc}(G)\). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) \(K_{1,4}\)-free split graphs, and it is polynomial time solvable for (i) trees and (ii) \(P_4\)-free graphs.     

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.     

Improving robustness of next-hop routing

A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following tree augmentation problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directed edge represents the unique next hop in the routing protocol. Our goal is to direct the undirected edges so that the resulting graph remains acyclic and the number of nodes with outdegree two or more is maximized. These nodes represent those with alternative next hops in their routing paths. We show that tree augmentation is NP-hard in general and present a simple \(\frac{1}{2}\)-approximation algorithm. We also study 3 special cases. We give exact polynomial-time algorithms for when the input spanning tree consists of exactly 2 directed paths or when the input graph has bounded treewidth. For planar graphs, we present a polynomial-time approximation scheme when the input tree is a breadth-first search tree. To the best of our knowledge, tree augmentation has not been previously studied.     

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.     

An improved parameterized algorithm for the <Emphasis Type="Italic">p</Emphasis>-cluster vertex deletion problem

In the p-Cluster Vertex Deletion problem, we are given a graph \(G=(V,E)\) and two parameters k and p, and the goal is to determine if there exists a subset X of at most k vertices such that the removal of X results in a graph consisting of exactly p disjoint maximal cliques. Let \(r=p/k\). In this paper, we design a branching algorithm with time complexity \(O(\alpha ^k+|V||E|)\), where \(\alpha \) depends on r and has a rough upper bound \(\min \{1.618^{1+r},2\}\). With a more precise analysis, we show that \(\alpha =1.28\cdot 3.57^{r}\) for \(r\le 0.219\); \(\alpha =(1-r)^{r-1}r^{-r}\) for \(0.219< r<1/2\); and \(\alpha =2\) for \(r\ge 1/2\), respectively. Our algorithm also works with the same time complexity for the variant that the number of clusters is at most p. Our result improves the previous best time complexity \(O^*(1.84^{p+k})\) and implies that for fixed p the problem can be solved as efficiently as Vertex Cover.     

OFDP: a distributed algorithm for finding disjoint paths with minimum total length in wireless sensor networks

This paper investigates the MINimum-length-\(k\)-Disjoint-Paths (MIN-\(k\)-DP) problem: in a sensor network, given two nodes \(s\) and \(t\), a positive integer \(k\), finding \(k\) (node) disjoint paths connecting \(s\) and \(t\) with minimum total length. An efficient distributed algorithm named Optimally-Finding-Disjoint-Paths (OFDP) is proposed for this problem. OFDP guarantees correctness and optimality, i.e., (1) it will find \(k\) disjoint paths if there exist \(k\) disjoint paths in the network or the maximum number of disjoint paths otherwise; (2) the disjoint paths it outputs do have minimum total length. To the best of our knowledge, OFDP is the first distributed algorithm that can solve the MIN-\(k\)-DP problem with correctness and optimality guarantee. Compared with the existing centralized algorithms which also guarantee correctness and optimality, OFDP is shown to be much more efficient by simulation results.     

Parameterized algorithms for min–max 2-cluster editing

For a given graph and an integer t, the Min–Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t. It has been shown that the problem can be solved in polynomial time for \(t<n/4\), where n is the number of vertices. In this paper, we design parameterized algorithms for different ranges of t. Let \(k=t-n/4\). We show that the problem is polynomial-time solvable when roughly \(k<\sqrt{n/32}\). When \(k\in o(n)\), we design a randomized and a deterministic algorithm with sub-exponential time parameterized complexity, i.e., the problem is in SUBEPT. We also show that the problem can be solved in \(O({2}^{n/r}\cdot n^2)\) time for \(k<n/12\) and in \(O(n^2\cdot 2^{3n/4+k})\) time for \(n/12\le k< n/4\), where \(r=2+\lfloor (n/4-3k-2)/(2k+1) \rfloor \ge 2\).     

The maximum flow problem with disjunctive constraints

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly $\mathcal{NP}$ -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly $\mathcal{NP}$ -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.     

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(\overline{3K_2}\)-free co-bipartite graphs and polynomially solvable on \(C_4\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most \(2^k\) vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most \(2^k\) vertices. Finally we determine all graphs of small complete width \(k\le 3\).     

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 \(\max \{b,4\}\).     

An $$O(|E(G)|^2)$$ algorithm for recognizing Pfaffian graphs of a type of bipartite graphs

A graph \(G=(V,E)\) with even number vertices is called Pfaffian if it has a Pfaffian orientation, namely it admits an orientation such that the number of edges of any M-alternating cycle which have the same direction as the traversal direction is odd for some perfect matching M of the graph G. In this paper, we obtain a necessary and sufficient condition of Pfaffian graphs in a type of bipartite graphs. Then, we design an \(O(|E(G)|^2)\) algorithm for recognizing Pfaffian graphs in this class and constructs a Pfaffian orientation if the graph is Pfaffian. The results improve and generalize some known results.     

Roman game domination number of a graph

The Roman game domination number of an undirected graph G is defined by the following game. Players \(\mathcal {A}\) and \(\mathcal {D}\) orient the edges of the graph G alternately, with \(\mathcal {D}\) playing first, until all edges are oriented. Player \(\mathcal {D}\) (frequently called Dominator) tries to minimize the Roman domination number of the resulting digraph, while player \(\mathcal {A}\) (Avoider) tries to maximize it. This game gives a unique number depending only on G, if we suppose that both \(\mathcal {A}\) and \(\mathcal {D}\) play according to their optimal strategies. This number is called the Roman game domination number of G and is denoted by \(\gamma _{Rg}(G)\). In this paper we initiate the study of the Roman game domination number of a graph and we establish some bounds on \(\gamma _{Rg}(G)\). We also determine the Roman game domination number for some classes of graphs.     

On the vertex cover $$P_3$$ problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.     

Complexity and inapproximability results for the Power Edge Set problem

We consider the single channel PMU placement problem called the Power Edge Set problem. In this variant of the PMU placement problem, (single channel) PMUs are placed on the edges of an electrical network. Such a PMU measures the current along the edge on which it is placed and the voltage at its two endpoints. The objective is to find the minimum placement of PMUs in the network that ensures its full observability, namely measurement of all the voltages and currents. We prove that PES is NP-hard to approximate within a factor (1.12)-\(\epsilon \), for any \(\epsilon > 0\). On the positive side we prove that PES problem is solvable in polynomial time for trees and grids.     

On the odd girth and the circular chromatic number of generalized Petersen graphs

A class \(\mathcal{G}\) of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph \(\mathrm {G} \in \mathcal{G}\) such that the girth (odd-girth) of \(\mathrm {G}\) is \(\ge g\). A girth-closed (odd-girth-closed) class \(\mathcal{G}\) of graphs is said to be pentagonal (odd-pentagonal) if there exists a positive integer \(g^*\) depending on \(\mathcal{G}\) such that any graph \(\mathrm {G} \in \mathcal{G}\) whose girth (odd-girth) is greater than \(g^*\) admits a homomorphism to the five cycle (i.e. is \(\mathrm {C}_{_{5}}\)-colourable). Although, the question “Is the class of simple 3-regular graphs pentagonal?” proposed by Ne?et?il (Taiwan J Math 3:381–423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has shown that there exists an odd-girth-closed subclass of simple 3-regular graphs which is not odd-pentagonal. In this article, motivated by the conjecture that the class of generalized Petersen graphs is odd-pentagonal, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using the combinatorial and number theoretic properties of this problem, we explicitly compute the odd girth of such graphs, showing that the class is odd-girth-closed. Also, we obtain upper and lower bounds for the circular chromatic number of these graphs, and as a consequence, we show that the subclass containing generalized Petersen graphs \(\mathrm {Pet}(n,k)\) for which either k is even, n is odd and \(n\mathop {\equiv }\limits ^{k-1}\pm 2\) or both n and k are odd and \(n\ge 5k\) is odd-pentagonal. This in particular shows the existence of nontrivial odd-pentagonal subclasses of 3-regular simple graphs.     

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.     

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.