Computing unique maximum matchings in $$$$ time for König–Egerváry graphs and unicyclic graphs

Let \(\alpha \left( G\right) \) denote the maximum size of an independent set of vertices and \(\mu \left( G\right) \) be the cardinality of a maximum matching in a graph \(G\). A matching saturating all the vertices is a perfect matching. If \(\alpha \left( G\right) +\mu \left( G\right) =\left| V(G)\right| \), then \(G\) is called a König–Egerváry graph. A graph is unicyclic if it is connected and has a unique cycle. It is known that a maximum matching can be found in \(O(m\cdot \sqrt{n})\) time for a graph with \(n\) vertices and \(m\) edges. Bartha (Proceedings of the 8th joint conference on mathematics and computer science, Komárno, Slovakia, 2010) conjectured that a unique perfect matching, if it exists, can be found in \(O(m)\) time. In this paper we validate this conjecture for König–Egerváry graphs and unicylic graphs. We propose a variation of Karp–Sipser leaf-removal algorithm (Karp and Sipser in Proceedings of the 22nd annual IEEE symposium on foundations of computer science, 364–375, 1981) , which ends with an empty graph if and only if the original graph is a König–Egerváry graph with a unique perfect matching (obtained as an output as well). We also show that a unicyclic non-bipartite graph \(G\) may have at most one perfect matching, and this is the case where \(G\) is a König–Egerváry graph.     

Maximum flows in generalized processing networks

Processing networks (cf. Koene in Minimal cost flow in processing networks: a primal approach, 1982) and manufacturing networks (cf. Fang and Qi in Optim Methods Softw 18:143–165, 2003) are well-studied extensions of traditional network flow problems that allow to model the decomposition or distillation of products in a manufacturing process. In these models, so called flow ratios \(\alpha _e \in [0,1]\) are assigned to all outgoing edges of special processing nodes. For each such special node, these flow ratios, which are required to sum up to one, determine the fraction of the total outgoing flow that flows through the respective edges. In this paper, we generalize processing networks to the case that these flow ratios only impose an upper bound on the respective fractions and, in particular, may sum up to more than one at each node. We show that a flow decomposition similar to the one for traditional network flows is possible and can be computed in strongly polynomial time. Moreover, we show that there exists a fully polynomial-time approximation scheme (FPTAS) for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic and we provide two exact algorithms with strongly polynomial running-time for the problem on series–parallel graphs. Finally, we study the case of integral flows and show that the problem becomes \({\mathcal {NP}}\)-hard to solve and approximate in this case.     

The maximum cardinality cut problem in co-bipartite chain graphs

A co-bipartite chain graph is a co-bipartite graph in which the neighborhoods of the vertices in each clique can be linearly ordered with respect to inclusion. It is known that the maximum cardinality cut problem (\({\textsc {MaxCut}}\)) is \({\textsc {NP}}{\text {-hard}}\) in co-bipartite graphs (Bodlaender and Jansen, Nordic J Comput 7(2000):14–31, 2000). We consider \({\textsc {MaxCut}}\) in co-bipartite chain graphs. We first consider the twin-free case and present an explicit solution. We then show that \({\textsc {MaxCut}}\) is polynomial time solvable in this graph class.     

Approximability of the subset sum reconfiguration problem

The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme, while the problem is APX-hard if we are given a conflict graph.     

The thickness of the complete multipartite graphs and the join of graphs

The thickness of a graph is the minimum number of planar spanning subgraphs into which the graph can be decomposed. It is known for relatively few classes of graphs, compared to other topological invariants, e.g., genus and crossing number. For the complete bipartite graphs, Beineke et al. (Proc Camb Philos Soc 60:1–5, 1964) gave the answer for most graphs in this family in 1964. In this paper, we derive formulas and bounds for the thickness of some complete k-partite graphs. And some properties for the thickness for the join of two graphs are also obtained.     

Approximation for vertex cover in $$\beta $$-conflict graphs

Conflict graph is a union of finite given sets of disjoint complete multipartite graphs. Vertex cover on this kind of graph is used first to model data inconsistency problems in database application. It is NP-complete if the number of given sets r is fixed, and can be approximated within \(2-\frac{1}{2^r}\) (Miao et al. in Proceedings of the 9th international conference on combinatorial optimization and applications, vol 9486. COCOA 2015, New York. Springer, New York, pp 395–408, 2015). This paper shows a better algorithm to improve the approximation for dense cases. If the ratio of vertex not belongs to any wheel complete multipartite graph is no more than \(\beta <1\), then our algorithm will provide a \((1+\beta +\frac{1-\beta }{k})\)-approximation, where k is a parameter related to degree distribution of wheel complete multipartite graph.     

Maximum cardinality neighbourly sets in quadrilateral free graphs

Neighbourly set of a graph is a subset of edges which either share an end point or are joined by an edge of that graph. The maximum cardinality neighbourly set problem is known to be NP-complete for general graphs. Mahdian (Discret Appl Math 118:239–248, 2002) proved that it is in polynomial time for quadrilateral-free graphs and proposed an \(O(n^{11})\) algorithm for the same, here n is the number of vertices in the graph, (along with a note that by a straightforward but lengthy argument it can be proved to be solvable in \(O(n^5)\) running time). In this paper we propose an \(O(n^2)\) time algorithm for finding a maximum cardinality neighbourly set in a quadrilateral-free graph.     

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.     

Shifting strategy for geometric graphs without?geometry

We give a simple framework which is an alternative to the celebrated and widely used shifting strategy of Hochbaum and Maass (J. ACM 32(1):103?C136, 1985) which has yielded efficient algorithms with good approximation bounds for numerous optimization problems in low-dimensional Euclidean space. Our framework does not require the input graph/metric to have a geometric realization??it only requires that the input graph satisfy some weak property referred to as growth boundedness. Growth bounded graphs form an important graph class that has been used to model wireless networks. We show how to apply the framework to obtain a polynomial time approximation scheme (PTAS) for the maximum (weighted) independent set problem on this important graph class; the problem is W[1]-complete. Via a more sophisticated application of our framework, we show how to obtain a PTAS for the maximum (weighted) independent set for intersection graphs of (low-dimensional) fat objects that are expressed without geometry. Erlebach et al. (SIAM J. Comput. 34(6):1302?C1323, 2005) and Chan (J. Algorithms 46(2):178?C189, 2003) independently gave a PTAS for maximum weighted independent set problem for intersection graphs of fat geometric objects, say ball graphs, which required a geometric representation of the input. Our result gives a positive answer to a question of Erlebach et al. (SIAM J. Comput. 34(6):1302?C1323, 2005) who asked if a PTAS for this problem can be obtained without access to a geometric representation.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.     

Competitive analysis of bi-directional non-preemptive conversion

In an online conversion problem a player is converting one asset into another, e.g. dollars to yen, based on a finite sequence of unknown future conversion rates. When a new rate is announced the player must decide how many dollars (yen) to convert to yen (dollars) according to this rate. Conversion is over when the last rate has appeared. The objective is to maximize terminal wealth after conversion. In uni-directional conversion dollars are converted to yen and in bi-directional conversion not only dollars to yen but also yen to dollars may be converted. If all current wealth has to be converted at one rate we call the problem non-preemptive; if parts of the current wealth can be converted at one rate we call it preemptive. We assume that lower and upper bounds on conversion rates are given. Uni-directional preemptive and non-preemptive and bi-directional preemptive conversion is investigated in El Yaniv et al. (Proceeding 33rd annual symposium on foundations of computer science, pp 327–333, 1992, Algorithmica 30:101–139, 2001). Their results for bi-directional preemptive conversion are improved by Dannoura and Sakurai (Inf Process Lett 66:27–33, 1998). The suggested improvement is conjectured not to be optimal for bi-directional preemptive conversion. There are no results for optimal bi-directional non-preemptive conversion. We investigate the problem of bi-directional non-preemptive online conversion. We present lower bounds, upper bounds and an optimal algorithm to solve the problem. Moreover, we prove that the algorithm of Dannoura and Sakurai 1998 is not optimal for bi-directional preemptive conversion.     

A solution to an open problem on lower against number in graphs

In Wang (Appl Anal Discret Math 4:207–218, 2010) the problem of finding a sharp lower bound on lower against number of a general graph is mentioned as an open question. We solve the problem by establishing a tight lower bound on lower against number of a general graph in terms of order and maximum degree.     

A solution to a conjecture on the generalized connectivity of graphs

The generalized k-connectivity \(\kappa _k(G)\) of a graph G was introduced by Chartrand et al. in (Bull Bombay Math Colloq 2:1–6, 1984), which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we consider the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. We give a confirmative solution to a conjecture raised by S. Li in Ph.D. Thesis (2012). We also give a polynomial-time algorithm to a problem of generalized k-edge-connectivity.     

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.     

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.     

A combinatorial proof for the circular chromatic number of Kneser graphs

Chen (J Combin Theory A 118(3):1062–1071, 2011) confirmed the Johnson–Holroyd–Stahl conjecture that the circular chromatic number of a Kneser graph is equal to its chromatic number. A shorter proof of this result was given by Chang et al. (J Combin Theory A 120:159–163, 2013). Both proofs were based on Fan’s lemma (Ann Math 56:431–437, 1952) in algebraic topology. In this article we give a further simplified proof of this result. Moreover, by specializing a constructive proof of Fan’s lemma by Prescott and Su (J Combin Theory A 111:257–265, 2005), our proof is self-contained and combinatorial.     

An Approximation Scheme for Bin Packing with Conflicts

In this paper we consider the following bin packing problem with conflicts. Given a set of items V = {1,..., n} with sizes s₁,..., s (0,1) and a conflict graph G = (V, E), we consider the problem to find a packing for the items into bins of size one such that adjacent items (j, j) E are assigned to different bins. The goal is to find an assignment with a minimum number of bins. This problem is a natural generalization of the classical bin packing problem.We propose an asymptotic approximation scheme for the bin packing problem with conflicts restricted to d-inductive graphs with constant d. This graph class contains trees, grid graphs, planar graphs and graphs with constant treewidth. The algorithm finds an assignment for the items such that the generated number of bins is within a factor of (1 + ) of optimal provided that the optimum number is sufficiently large. The running time of the algorithm is polynomial both in n and in .     

Two-step values for games with two-level communication structure

TU games with two-level communication structure, in which a two-level communication structure relates fundamentally to the given coalition structure and consists of a communication graph on the collection of the a priori unions in the coalition structure, as well as a collection of communication graphs within each union, are considered. For such games we introduce two families of two-step values inspired by the two-step procedures staying behind the Owen value (Owen, in: Henn, Moeschlin (eds) Essays in mathematical economics and game theory, Springer, Berlin, pp 76–88, 1977) and the two-step Shapley value (Kamijo in Int Game Theory Rev 11:207–214, 2009) for games with coalition structure. Our approach is based on the unified treatment of several component efficient values for games with communication structure and it generates two-stage solution concepts that apply component efficient values for games with communication structure on both distribution levels. Comparable axiomatic characterizations are provided.     

Note on power propagation time and lower bounds for the power domination number

We present a counterexample to a lower bound for the power domination number given in Liao (J Comb Optim 31:725–742, 2016). We also define the power propagation time, using the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al. (Discrete Appl Math 160:1994–2005, 2012).