Any Maximal Planar Graph with Only One Separating Triangle is Hamiltonian

A graph is hamiltonian if it has a hamiltonian cycle. It is well-known that Tutte proved that any 4-connected planar graph is hamiltonian. It is also well-known that the problem of determining whether a 3-connected planar graph is hamiltonian is NP-complete. In particular, Chvátal and Wigderson had independently shown that the problem of determining whether a maximal planar graph is hamiltonian is NP-complete. A classical theorem of Whitney says that any maximal planar graph with no separating triangles is hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Note that if a planar graph has separating triangles, then it can not be 4-connected and therefore Tutte's result can not be applied. In this paper, we shall prove that any maximal planar graph with only one separating triangle is still hamiltonian.     

The optimum traversal of a graph

For a given graph (network) having costs [c_ij] associated with its links, the present paper examines the problem of finding a cycle which traverses every link of the graph at least once, and which incurs the minimum cost of traversal. This problem (called thegraph traversal problem, or theChinese postman problem [9]) can be formulated in ways analogous to those used for the well-known travelling salesman problem, and using this apparent similarity, Bellman and Cooke [1] have produced a dynamic programming formulation. This method of solution of the graph traversal problem requires computational times which increase exponentially with the number of links in the graph. Approximately the same rate of increase of computational effort with problem size would result by any other method adapting a travelling salesman algorithm to the present problem.This paper describes an efficient algorithm for the optimal solution of the graph traversal problem based on the matching method of Edmonds [5, 6]. The computational time requirements of this algorithm increase as a low order (2 or 3) power of the number of links in the graph. Computational results are given for graphs of up to 50 vertices and 125 links.The paper then discusses a generalised version of the graph traversal problem, where not one but a number of cycles are required to traverse the graph. In this case each link has (in addition to its cost) a quantity q_ij associated with it, and the sum of the quantities of the links in any one cycle must be less than a given amount representing the cycle capacity. A heuristic algorithm for the solution of this problem is given. The algorithm is based on the optimal algorithm for the single-cycle graph traversal problem and is shown to produce near-optimal results.There is a large number of possible applications where graph traversal problems arise. These applications include: the spraying of roads with salt-grit to prevent ice formation, the inspection of electric power lines, gas, or oil pipelines for faults, the delivery of letter post, etc.     

Generalized median graphs and applications

We study the so-called Generalized Median graph problem where the task is to construct a prototype (i.e., a ‘model’) from an input set of graphs. While our primary motivation comes from an important biological imaging application, the problem effectively captures many vision (e.g., object recognition) and learning problems, where graphs are increasingly being adopted as a powerful representation tool. Existing techniques for his problem are evolutionary search based; in this paper, we propose a polynomial time algorithm based on a linear programming formulation. We propose an additional algorithm based on a bi-level method to obtain solutions arbitrarily close to the optimal in (worst case) non-polynomial time. Within this new framework, one can optimize edit distance functions that capture similarity by considering vertex labels as well as he graph structure simultaneously. We first discuss experimental evaluations in context of molecular image analysis problems—he methods will provide the basis for building a topological map of all 23 pairs of the human chromosome. Later, we include (a) applications to other biomedical problems and (b) evaluations on a public pattern recognition graph database. This work was supported by NSF grants CCF-0546509, IIS-0713489, and NIH grant GM 072131-23. The second author was also supported in part by the Department of Biostatistics and Medical Informatics, UW-Madison and UW ICTR, funded through an NIH Clinical and Translational Science Award (CTSA), grant number 1 UL1 RR025011.     

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.     

Degree-constrained orientations of embedded graphs

We investigate the problem of orienting the edges of an embedded graph in such a way that the resulting digraph fulfills given in-degree specifications both for the vertices and for the faces of the embedding. This primal-dual orientation problem was first proposed by Frank for the case of planar graphs, in conjunction with the question for a good characterization of the existence of such orientations. We answer this question by showing that a feasible orientation of a planar embedding, if it exists, can be constructed by combining certain parts of a primally feasible orientation and a dually feasible orientation. For the general case of arbitrary embeddings, we show that the number of feasible orientations is bounded by \(2^{2g}\), where \(g\) is the genus of the embedding. Our proof also yields a fixed-parameter algorithm for determining all feasible orientations parameterized by the genus. In contrast to these positive results, however, we also show that the problem becomes \(N\!P\)-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.     

A “maximum node clustering” problem

In this note we introduce a graph problem, called Maximum Node Clustering (MNC). We prove that the problem (which is easily shown to be strongly NP-complete) can be approximated in polynomial time within a ratio arbitrarily close to 2. For the special case where the graph is a tree, the problem is NP-complete in the ordinary sense; for this case we present a pseudopolynomial algorithm based on dynamic programming, and a related Fully Polynomial Time Approximation Scheme (FPTAS). Also, the tree case is shown to be exactly solvable in time, where n is the number of nodes.     

Order consolidation for hierarchical product lines

We consider the batch production of hierarchical product lines in raw material industry where the whole or parts of multiple customer orders may be consolidated and processed in the same batch if their product specifications are compatible. The objective of the problem is to find maximum possible number of batches completely filled up to their capacity. The compatibility relationship among product specifications is represented by a graph called the compatibility graph. If the compatibility graph is an arbitrary graph, the problem is proven to be NP-hard and belongs to Max SNP-hard class. We develop an optimum algorithm for an important subclass of the problem where the graph is a quasi-threshold graph which in fact is the case for producing hierarchical product lines that are often found in raw materials industry.     

Augmenting weighted graphs to establish directed point-to-point connectivity

We consider an augmentation problem on undirected and directed graphs, where given a directed (an undirected) graph G and p pairs of vertices \(P=\left\{ {\left( {s_1 ,t_1 } \right) ,\ldots ,\left( {s_p ,t_p } \right) } \right\} \), one has to find the minimum weight set of arcs (edges) to be added to the graph so that the resulting graph has (can be oriented to have) directed paths between the specified pairs of vertices. In the undirected case, we present an FPT-algorithm with respect to the number of new edges. Also, we have implemented and evaluated the algorithm on some real-world networks to show its efficiency in decreasing the size of input graphs and converting them to much smaller kernels. In the directed case, we consider the complexity of the problem with respect to the various parameters and present some parameterized algorithms and parameterized complexity results for it.     

A simplex like approach based on star sets for recognizing convex-QP adverse graphs

A graph \(G\) with convex-\(QP\) stability number (or simply a convex-\(QP\) graph) is a graph for which the stability number is equal to the optimal value of a convex quadratic program, say \(P(G)\). There are polynomial-time procedures to recognize convex-\(QP\) graphs, except when the graph \(G\) is adverse or contains an adverse subgraph (that is, a non complete graph, without isolated vertices, such that the least eigenvalue of its adjacency matrix and the optimal value of \(P(G)\) are both integer and none of them changes when the neighborhood of any vertex of \(G\) is deleted). In this paper, from a characterization of convex-\(QP\) graphs based on star sets associated to the least eigenvalue of its adjacency matrix, a simplex-like algorithm for the recognition of convex-\(QP\) adverse graphs is introduced.     

A heuristic for facilities layout planning

A new matrix representation of a planar graph and its dual is presented. This is then used to implement a heuristic for facilities layout planning.     

Computing the <Emphasis Type="Italic">k</Emphasis>-resilience of a synchronized multi-robot system

The vertex arboricity va(G) of a graph G is the minimum number of colors the vertices can be colored so that each color class induces a forest. It was known that \(va(G)\le 3\) for every planar graph G. In this paper, we prove that \(va(G)\le 2\) if G is a planar graph without intersecting 5-cycles.     

Column generation approach for the point-feature cartographic label placement problem

This paper proposes a column generation approach for the Point-Feature Cartographic Label Placement problem (PFCLP). The column generation is based on a Lagrangean relaxation with clusters proposed for problems modeled by conflict graphs. The PFCLP can be represented by a conflict graph where vertices are positions for each label and edges are potential overlaps between labels (vertices). The conflict graph is decomposed into clusters forming a block diagonal matrix with coupling constraints that is known as a restricted master problem (RMP) in a Dantzig-Wolfe decomposition context. The clusters’ sub-problems are similar to the PFCLP and are used to generate new improved columns to RMP. This approach was tested on PFCLP instances presented in the literature providing in reasonable times better solutions than all those known and determining optimal solutions for some difficult large-scale instances.     

Online graph exploration algorithms for cycles and trees by multiple searchers

This paper deals with online graph exploration problems by multiple searchers. The information on the graph is given online. As the exploration proceeds, searchers gain more information on the graph. Assuming an appropriate communication model among searchers, searchers can share the information about the environment. Thus, a searcher must decide which vertex to visit next based on the partial information on the graph gained so far by searchers. We assume that all searchers initially start the exploration at the origin vertex, and the goal is that each vertex is visited by at least one searcher and all searchers finally return to the origin vertex. The objective is to minimize the time when the goal is achieved. We study the case of cycles and trees. For the former, we give an optimal online exploration algorithm in terms of competitive ratio, and for the latter, we also give an online exploration algorithm which is optimal among greedy algorithms.     

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 .     

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.     

The capture time of a planar graph

This paper examines the capture time of a planar graph in a variant of the pursuit-evasion games, called cops and robbers game. Since any planar graph is 3-cop-win, we study the capture time of a planar graph G of n vertices using three cops, which is denoted by \(capt_3(G)\). We present a new capture strategy and show that \(capt_3(G) \le 2n\). This is the first result on \(capt_3(G)\).     

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.     

Approximation of knapsack problems with conflict and forcing graphs

We study the classical 0–1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570–581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. The PTAS is obtained by expanding a general result of Demaine et al. (Proceedings of 46th annual IEEE symposium on foundations of computer science, FOCS 2005, pp 637–646, 2005). The knapsack problem with forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.     

Minimum entropy coloring

We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that, for every constant ε>0, it is NP-hard to find a coloring whose entropy is within (1−ε)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem. S. Fiorini acknowledges the support from the Fonds National de la Recherche Scientifique and GERAD-HEC Montréal. G. Joret is a F.R.S.-FNRS Research Fellow.     

Total coloring of planar graphs without adjacent short cycles

In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let \(G=(V,E)\) be a graph. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with \(\varDelta \ge 8\). We proved that if for every vertex \(v\in V\), there exists two integers \(i_v,j_v\in \{3,4,5,6,7\}\) such that v is not incident with adjacent \(i_v\)-cycles and \(j_v\)-cycles, then the total chromatic number of graph G is \(\varDelta +1\).