A note on the Hamiltonian Circuit Problem on directed path graphs

Chordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

The graph bipartization problem, arising from via minimization in VLSI design and related areas, consists in finding a vertex subset [Formula: see text] of graph [Formula: see text] such that the induced subgraph [Formula: see text] is bipartite and [Formula: see text] is maximized. The problem has been proved to be NP-hard even for planar graphs and cubic graphs. On the other hand, the study of polynomial-time algorithms for typical graph classes is significant in both theoretical and applied aspects. This paper focuses on several intersection graph classes, such as line graphs, circular-arc graphs, and directed path graphs. For the line graphs, we show the NP-hardness results in general and present the polynomial-time algorithms for special cases. For circular-arc graphs and directed path graphs, we propose algorithms that improve on the previously known ones.

Suppose G is a connected graph and T a spanning tree of G. A vertex v ε V(G) is said to be a degree‐preserving vertex if its degree in T is the same as its degree in G. The degree‐preserving spanning tree problem is to find a spanning tree T of a connected graph G such that the number of degree‐preserving vertices is maximized. The purpose of this article is to provide an O(m.α(m,n))‐time algorithm for the degree‐preserving spanning tree problem in strongly chordal graphs, where α is the inverse of Ackermann's function. Furthermore, we present an O(m + n)‐time algorithm in directed path graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Molecular computing has proved its possibility to solve weighted graph problem such as Hamiltonian Path Problem (HPP), Traveling Salesman Problem (TSP) and the Shortest Path Problem (SPP). Normally, in molecular computation, the DNA sequences used for the computation should be critically designed in order to reduce error that could occur during computation. In the previous paper, we have proposed a readout method tailored specifically to HPP in DNA Computing using real-time PCR for output visualization. Six nodes of HPP was considered. Based on the example instance, the method requires 11 oligonucleotides, where 6 oligonucleotides are for the nodes and 5 oligonucleotides are for the edges. Three TaqMan probes and five primers are required as well. In this study, a procedure for DNA sequence design is presented in order to obtain good sequences for those nodes, primers, and probes. The experiment is done based on the generated DNA sequences and the Hamiltonian Path can be determined successfully.

We present a genetic-based approach to solve SAT problem and NP-complete problems. The main idea of the approach presented here is to exploit the fact that, although all NP-complete problems are equally difficult in a general computational sense, some have much better genetic representations than others, leading to much more successful use of genetic-based algorithm on some NP-complete problems than on others. Since any NP-complete problem can be mapped into any other one in polynomial time by a transformation, the approach described here consists of identifying and finding a canonical or generic NP-complete problem on which genetic algorithm work well, and solving other NP-complete problems indirectly by translating them onto the canonical problem. We presented some initial results where we have the Boolean Satisfiability Problem (SAT) as a canonical problem, and results on Hamiltonian Circuit problem which represent a family of NP-complete problems, it can be solved efficiently by mapping them first onto SAT problems.KeywordsSATGenetic AlgorithmNP-complete ProblemHamiltonian Circuit

A heuristic search algorithm is given that determines and resolves the Hamiltonian circuit problem in directed graphs. The heuristic information of each vertex is a set composed of its possible path length values from the starting vertex, which is obtained by the path length extension algorithm. A detailed analysis of algorithm examples shows that the heuristic algorithm can greatly reduce the number of processing nodes compared to the backtracking algorithm without information; meanwhile, the algorithm’s time complexity is related to the number of loops in the directed graph. If a directed graph G(V, E) only contains loops that contain the starting point, then the Hamiltonian problem of G can be determined in polynomial time. The algorithm is implemented using C++ and Python programming, and all of the test graphs are generated by the random graph algorithm. The experiments show that the heuristic algorithm can reduce the number of extended nodes by at least 10% in directed random graphs with no or only a few loops. This heuristic algorithm can save time in applications that perform tasks while extending nodes because it reduce the number of extended nodes.

Computing residual connectedness reliability for restricted networks

k-tuple domination in graphs

The complexity of the locally connected spanning tree problem

The concept of a line digraph is generalized to that of a directed path graph. The directed path graph $\forw P_k(D)$ of a digraph $D$ is obtained by representing the directed paths on $k$ vertices of $D$ by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in $D$ form a directed path on $k+1$ vertices or form a directed cycle on $k$ vertices in $D$. In this introductory paper several properties of $\forw P_3(D)$ are studied, in particular with respect to isomorphism and traversability. In our main results, we characterize all digraphs $D$ with $\forw P_3(D)\cong D$, we show that $\forw P_3(D_1)\cong\forw P_3(D_2)$ ``almost always'' implies $D_1\cong D_2$, and we characterize all digraphs with Eulerian or Hamiltonian $\forw P_3$-graphs.

A faster algorithm to recognize undirected path graphs

Clique graphs of chordal and path graphs are characterized. A special class of graphs named expanded trees is discussed. It consists of a subclass of disk-Helly graphs. It is shown that the clique graph of every chordal (hence path) graph is an expanded tree. In addition, every expanded tree is the clique graph of some path (hence chordal) graph. Different characterizations of expanded trees are described, leading to a polynomial time algorithm for recognizing them.

Let G V , E be a simple undirected complete graph with vertex and edge sets V and E , respectively. In this paper, we consider the degree-constrained k -minimum spanning tree (DC k MST) problem which consists of finding a minimum cost subtree of G formed with at least k vertices of V where the degree of each vertex is less than or equal to an integer value d ≤ k − 2 . In particular, in this paper, we consider degree values of d ∈ 2,3 . Notice that DC k MST generalizes both the classical degree-constrained and k -minimum spanning tree problems simultaneously. In particular, when d = 2 , it reduces to a k -Hamiltonian path problem. Application domains where DC k MST can be adapted or directly utilized include backbone network structures in telecommunications, facility location, and transportation networks, to name a few. It is easy to see from the literature that the DC k MST problem has not been studied in depth so far. Thus, our main contributions in this paper can be highlighted as follows. We propose three mixed-integer linear programming (MILP) models for the DC k MST problem and derive for each one an equivalent counterpart by using the handshaking lemma. Then, we further propose ant colony optimization (ACO) and variable neighborhood search (VNS) algorithms. Each proposed ACO and VNS method is also compared with another variant of it which is obtained while embedding a Q-learning strategy. We also propose a pure Q-learning algorithm that is competitive with the ACO ones. Finally, we conduct substantial numerical experiments using benchmark input graph instances from TSPLIB and randomly generated ones with uniform and Euclidean distance costs with up to 400 nodes. Our numerical results indicate that the proposed models and algorithms allow obtaining optimal and near-optimal solutions, respectively. Moreover, we report better solutions than CPLEX for the large-size instances. Ultimately, the empirical evidence shows that the proposed Q-learning strategies can bring considerable improvements.

Many practical problems (frequency assignement, register allocation, timetables) may be formulated as graph (vertex-)coloring problems, but finding solutions for them is difficult as long as they are treated in the most general case (where the graph is arbitrary), since vertex coloring has been proved to be NP-complete. The problem becomes much easier to solve if the graph resulting from the modelisation of the practical application belongs to some particular class of graphs, for which solutions to the problem are known. Meyniel graphs form such a class (a fast coloring algorithm can be found in [9]), for which an efficient recognizing algorithm is therefore needed. A graph G=(V,E) is said to be a Meyniel graph if every odd cycle of G on at least five vertices contains at least two chords. Meyniel graphs generalize both i-triangulated and parity graphs, two well known classes of perfect graphs that will be present in our paper in Section 7. In [2], Burlet and Fonlupt propose a characterization of Meyniel graphs which relies on the following property: the class of Meyniel graphs may be obtained from some basic Meyniel graphs using a binary operation called amalgam. Besides the theoretical interest of this result, a practical interest arises because of the polynomial recognition algorithm which can be obtained. Unfortunately, it is quite expensive to verify if a given graph is the amalgam of two graphs (therefore the complexity of the whole algorithm is in O(n7)), and this supports the idea that a new point of view is needed to find a more efficient algorithm. Our approach of Meyniel graphs will be directed through the search of a general structure. Intuitively, a Meyniel graph either will be simple (i.e. with no hole or domino), or will have a skeleton around which the rest of the graph will be regularly organized. As suggested, the first type of Meyniel graphs is simple to identify. For the second type, a deeper analysis is necessary; it yields a characterization theorem, which is used to deduce the O(m2+mn) recognition algorithm.

Not being (super)thin or solid is hard: A study of grid Hamiltonicity