A complexity dichotomy and a new boundary class for the dominating set problem

We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

Max-coloring paths: tight bounds and extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G=(V,E) with vertex weights w such that $\sum_{i=1}^{k}\max_{v\in C_{i}}w(v_{i})$ is minimized, where C ₁,??,C _k are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V|+time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of ??(|V|log?|V|) in the algebraic computation tree model.     

Transversal Graphs for Partially Ordered Sets: Sequencing, Merging and Scheduling Problems

This paper introduces an approach to solving combinatorial optimization problems on partially ordered sets by the reduction to searching source-sink paths in the related transversal graphs. Different techniques are demonstrated in application to finding consistent supersequences, merging partially ordered sets, and machine scheduling with precedence constraints. Extending the approach to labeled partially ordered sets we also propose a solution for the smallest superplan problem and show its equivalence to the well studied coarsest regular refinement problem. For partially ordered sets of a fixed width the number of vertices in their transversal graphs is polynomial, so the reduction allows us easily to establish that many related problems are solvable in polynomial or pseudopolynomial time. For example, we establish that the longest consistent supersequence problem with a fixed number of given strings can be solved in polynomial time, and that the precedence-constrained release-date maximum- or total-cost preemptive or nonpreemptive job-shop scheduling problem with a fixed number of jobs can be solved in pseudopolynomial time. We also show that transversal graphs can be used to generalize and strengthen similar results obtained earlier by dynamic programming.     

An optimal square coloring of planar graphs

The square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors. In 1977, Wegner posed a conjecture on square coloring of planar graphs. The conjecture is still open. In this paper, we prove that Wegner’s conjecture is true for planar graphs with girth at least?6.     

Compact labelings for efficient first-order model-checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is nicely locally clique-width-decomposable. This notion generalizes that of a nicely locally tree-decomposable class. The graphs of such classes can be covered by graphs of bounded clique-width with limited overlaps. We also consider such labelings for bounded first-order formulas on graph classes of bounded expansion. Some of these results are extended to counting queries.     

Visual cryptography on graphs

In this paper, we consider a new visual cryptography scheme that allows for sharing of multiple secret images on graphs: we are given an arbitrary graph (V,E) where every node and every edge are assigned an arbitrary image. Images on the vertices are “public” and images on the edges are “secret”. The problem that we are considering is how to make a construction such that when the encoded images of two adjacent vertices are printed on transparencies and overlapped, the secret image corresponding to the edge is revealed. We define the most stringent security guarantees for this problem (perfect secrecy) and show a general construction for all graphs where the cost (in terms of pixel expansion and contrast of the images) is proportional to the chromatic number of the cube of the underlying graph. For the case of bounded degree graphs, this gives us constant-factor pixel expansion and contrast. This compares favorably to previous works, where pixel expansion and contrast are proportional to the number of images.     

A refined algorithm for maximum independent set in degree-4 graphs

The maximum independent set problem is one of the most important problems in theoretical analysis on time and space complexities of exact algorithms. Theoretical improvement on upper bounds on time complexity to solve this problem in low-degree graphs can lead to an improvement on that to the problem in general graphs. In this paper, we derive an upper bound \(O^*(1.1376^n)\) on the time complexity of a polynomial-space algorithm that solves the maximum independent set problem in an n-vertex graph with degree bounded by 4, improving all previous upper bounds on the time complexity of exact algorithms to this problem. Our algorithm is a branch-and-reduce algorithm and analyzed by using the measure-and-conquer method. To make an amortized analysis of the running time bound, we use an idea of “shift” to save some decrease of the measure from good branches to bad branches. Our algorithm first deals with small vertex cuts and vertices of degree \({\ge }5\), which may be created in our algorithm even if the input graph has maximum degree 4, then eliminates cycles of length 3 and 4 containing degree-4 vertices, and finally branches on degree-4 vertices. We invoke an exact algorithm for this problem in graphs with maximum degree 3 directly when the graph has no vertices of degree \({\ge }4\). Branching on degree-4 vertices on special local structures will be the bottleneck case, and we carefully design rules of choosing degree-4 vertices to branch on so that the resulting instances after branching decrease the measure effectively in the next step.     

Vertex and Tree Arboricities of Graphs

This paper studies the following variations of arboricity of graphs. The vertex (respectively, tree) arboricity of a graph G is the minimum number va(G) (respectively, ta(G)) of subsets into which the vertices of G can be partitioned so that each subset induces a forest (respectively, tree). This paper studies the vertex and the tree arboricities on various classes of graphs for exact values, algorithms, bounds, hamiltonicity and NP-completeness. The graphs investigated in this paper include block-cactus graphs, series-parallel graphs, cographs and planar graphs.     

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.     

Median problems with positive and negative weights on cycles and cacti

This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest facility. For the MWD model it is shown that there exists a finite set of points in the graph which contains the locations of facilities in an optimal solution. Furthermore, algorithms for both models for the 2-median problem on a cycle are developed. The algorithm for the MWD model runs in linear time, whereas the algorithm for the WMD model has a time complexity of  O(n²)\mathcal{O}(n^{2}) .     

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 \(k\le 3\).     

Partitioning 2-edge-colored complete multipartite graphs into monochromatic cycles, paths and trees

In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP-complete for complete multipartite graphs and the 2-PGMT problem is NP-complete for bipartite graphs. This also implies that all these three problems are NP-complete for general graphs, which solves a question proposed by the authors in a previous paper. Nevertheless, we show that the 2-PGMT problem can be solved in polynomial time for complete multipartite graphs. Research supported by NSFC.     

Linear time construction of 5-phylogenetic roots for tree chordal graphs

Inspired by phylogenetic tree construction in computational biology, Lin et al. (The 11th Annual International Symposium on Algorithms and Computation (ISAAC 2000), pp. 539–551, 2000) introduced the notion of a k -phylogenetic root. A k-phylogenetic root of a graph G is a tree T such that the leaves of T are the vertices of G, two vertices are adjacent in G precisely if they are within distance k in T, and all non-leaf vertices of T have degree at least three. The k-phylogenetic root problem is to decide whether such a tree T exists for a given graph G. In addition to introducing this problem, Lin et al. designed linear time constructive algorithms for k≤4, while left the problem open for k≥5. In this paper, we partially fill this hole by giving a linear time constructive algorithm to decide whether a given tree chordal graph has a 5-phylogenetic root; this is the largest class of graphs known to have such a construction.     

The max quasi-independent set problem

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f-max quasi-independent set, consists of, given a graph and a non-decreasing function f, finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f(|Q|). For this problem, we show an exact solution method that runs within time O^*(2^{\fracd-27/23d+1n})O^{*}(2^{\frac{d-27/23}{d+1}n}) on graphs of average degree bounded by d. For the most specifically defined γ-max quasi-independent set and k-max quasi-independent set problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.     

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.     

Integer programming approach to static monopolies in graphs

A subset M of vertices of a graph is called a static monopoly, if any vertex v outside M has at least \(\lceil \tfrac{1 }{2}\deg (v)\rceil \) neighbors in M. The minimum static monopoly problem has been extensively studied in graph theoretical context. We study this problem from an integer programming point of view for the first time and give a linear formulation for it. We study the facial structure of the corresponding polytope, classify facet defining inequalities of the integer programming formulation and introduce some families of valid inequalities. We show that in the presence of a vertex cut or an edge cut in the graph, the problem can be solved more efficiently by adding some strong valid inequalities. An algorithm is given that solves the minimum monopoly problem in trees and cactus graphs in linear time. We test our methods by performing several experiments on randomly generated graphs. A software package is introduced that solves the minimum monopoly problem using open source integer linear programming solvers.     

The density maximization problem in graphs

We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G=(V,E) with edge weights w _e ∈? and edge lengths ? _e ∈? for e∈E we define the density of a pattern subgraph H=(V′,E′)?G as the ratio ?(H)=∑ _e∈E′ w _e /∑ _e∈E′ ? _e . We consider the problem of computing a maximum density pattern H under various additional constraints. In doing so, we compute a single Pareto-optimal solution with the best weight per cost ratio subject to additional constraints further narrowing down feasible solutions for the underlying bi-objective network construction problem. First, we consider the problem of computing a maximum density pattern with weight at least W and length at most L in a host G. We call this problem the biconstrained density maximization problem. This problem can be interpreted in terms of maximizing the return on investment for network construction problems in the presence of a limited budget and a target profit. We consider this problem for different classes of hosts and patterns. We show that it is NP-hard, even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty. Second, we consider the maximum density subgraph problem under structural constraints on the vertex set that is used by the patterns. While a maximum density perfect matching can be computed efficiently in general graphs, the maximum density Steiner-subgraph problem, which requires a subset of the vertices in any feasible solution, is NP-hard and unlikely to admit a constant-factor approximation. When parameterized by the number of vertices of the pattern, this problem is W[1]-hard in general graphs. On the other hand, it is FPT on planar graphs if there is no constraint on the pattern and on general graphs if the pattern is a path.     

The matching extension problem in general graphs is co-NP-complete

A simple connected graph G with 2n vertices is said to be k-extendable for an integer k with \(0<k<n\) if G contains a perfect matching and every matching of cardinality k in G is a subset of some perfect matching. Lakhal and Litzler (Inf Process Lett 65(1):11–16, 1998) discovered a polynomial algorithm that decides whether a bipartite graph is k-extendable. For general graphs, however, it has been an open problem whether there exists a polynomial algorithm. The new result presented in this paper is that the extendability problem is co-NP-complete.     

Fast searching games on graphs

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all the intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to compute the fast search number of Halin graphs and their extensions. We present a quadratic time algorithm to compute the fast search number of cubic graphs.