Some Characterizations and NP-Complete Problems for Power Cordial Graphs

The Maximum Weight Stable Set (MWS) Problem is one of the fundamental algorithmic problems in graphs. It is NP-complete in general, and it has polynomial time solutions on many particular graph classes, some of them defined by forbidden subgraphs. A classical example is the result of Minty that the MWS problem is polynomially solvable for the class of claw-free graphs. The complexity of the MWS problem is unknown for the class of P5-free graphs. We give a survey on recently obtained efficient algorithms for the MWS problem for several graph classes defined by forbidden subgraphs where the algorithm avoids to recognize whether the (arbitrary) input graph is in the class i.e. the output is either a correct solution of the MWS problem or the fact that the input graph is not in the class. Such algorithms were called robust by Spinrad. The algorithms use the concepts of modular decomposition and of clique width of graphs.

Graph properties checkable in linear time in the number of vertices

Preface. 1 Introduction. 1.1 Two examples of combinatorial optimization. 1.2 Why study combinatorial optimization using statistical physics? 1.3 Textbooks. Bibliography. 2 Algorithms. 2.1 Pidgin Algol. 2.2 Iteration and recursion. 2.3 Divide-and-conquer. 2.4 Dynamic programming. 2.5 Backtracking. Bibliography. 3 Introduction to graphs. 3.1 Basic concepts and graph problems. 3.2 Basic graph algorithms. 3.3 Random graphs. Bibliography. 4 Introduction to complexity theory. 4.1 Turing machines. 4.2 Church's thesis. 4.3 Languages. 4.4 The halting problem. 4.5 Class P. 4.6 Class NP. 4.7 Definition of NP-completeness. 4.8 NP-complete problems. 4.9 Worst-case vs. typical-case complexity. Bibliography. 5 Statistical mechanics of the Ising model. 5.1 Phase transitions. 5.2 Some general notes on statistical mechanics. 5.3 The Curie-Weiss model of a ferromagnet. 5.4 The Ising model on a random graph. Bibliography. 6 Algorithms and numerical results for vertex covers. 6.1 Definitions. 6.2 Heuristic algorithms. 6.3 Branch-and-bound algorithm. 6.4 Results: Covering random graphs. 6.5 The leaf-removal algorithm. 6.6 Monte Carlo simulations. 6.7 Backbone. 6.8 Clustering of minimum vertex covers. Bibliography. 7 Statistical mechanics of vertex covers on a random graph. 7.1 Introduction. 7.2 The first-moment bound. 7.3 The hard-core lattice gas. 7.4 Replica approach. Bibliography. 8 The dynamics of vertex-cover algorithms. 8.1 The typical-case solution time of a complete algorithm. 8.2 The dynamics of generalized leaf-removal algorithms. 8.3 Random restart algorithms. Bibliography. 9 Towards new, statistical-mechanics motivated algorithms. 9.1 The cavity graph. 9.2 Warning propagation. 9.3 Belief propagation. 9.4 Survey propagation. 9.5 Numerical experiments on random graphs. Bibliography. 10 The satisfiability problem. 10.1 SAT algorithms. 10.2 Phase transitions in random K-SAT. 10.3 Typical-case dynamics of RandomWalkSAT. 10.4 Message-passing algorithms for SAT. Bibliography. 11 Optimization problems in physics. 11.1 Monte Carlo optimization. 11.2 Hysteric optimization. 11.3 Genetic algorithms. 11.4 Shortest paths and polymers in random media. 11.5 Maximum flows and random-field systems. 11.6 Submodular functions and free energy of Potts model. 11.7 Matchings and spin glasses. Bibliography. Index.

On knot-free vertex deletion: Fine-grained parameterized complexity analysis of a deadlock resolution graph problem

Many graph parameters are the optimal value of an objective function over selected subsets S of vertices with some constraint on how many selected neighbors vertices in S, and vertices not in S, can have. Classic examples are minimum dominating set and maximum independent set. We give a characterization of these graph parameters that unifies their definitions, facilitates their common algorithmic treatment and allows for their uniform complexity classification. This characterization provides the basis for a taxonomy of domination-type and independence-type problems. We investigate the computational complexity of problems within this taxonomy, identify classes of NP-complete problems and classes of problems solvable in polynomial time.

We study in this paper how the computational behaviour of NIP-complete graph problems is influenced by imposing a bound on the bandwidth. We show: (i) A large number of problems are considerably simpler for graphs of small bandwidth. On the other hand, there are problems which remain NIP-complete even for graphs of bandwidth 3. (ii) A large number of problems are equivalent under reductions which preserve bandwidth. All these problems are complete under bandwidth preserving reductions for a class RPP, which is defined by nondeterministic Turing machines operating with some space bound and simultaneous polynomial time. This indicates, because of earlier results, that "bandwidth" plays the same role for graph problems as "maximal number" does for number problems. KeywordsSchedule ProblemPolynomial TimeTuring MachineconjUnctive Normal FormGraph ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two cycles and the union any two paths are cubic cordial graphs. Also, we study the cubic cordiality for the union of any cycle with a path.

Generation of all maximal independent sets (MIS) is a NP-complete problem for general graphs. In this paper, using a particular geometric property of trapezoid graph, an efficient algorithm is designed which enables to generate all maximal independent sets for a trapezoid graph in time, where [mbar] and a are respectively the number of edges of the complement graph of the given graph and the number of generated maximal independent sets. There is no prior algorithm available to solve this problem of finding all MIS for trapezoid graphs.

An important property of graph which concerns their applications to networks is Hamiltonicity. Determining if a graph is hamiltonian is a NP-complete problem and no satisfactory characterization exists. Nevertheless, many sufficient conditions were found, usually expressed in terms of degree, connectivity, density, toughness, independent set, regularity and forbidden subgraphs. In 1973, Goodman and Hedetniemi gave two alternative sufficient conditions uniquely based on the possibility to decompose in some ways the graph into cliques. We first generalise one of these two conditions. We give a new clique decomposition condition which proves the hamiltonicity of a broader class of graphs. Then, we present a polynomial algorithm which decides the existence of such a decomposition for simplicial-connected graphs. A graph is simplicial-connected if every vertex is connected by a walk to a simplicial one1. In particular, each connected graph containing a simplicial vertex is simplicial-connected, and so is every chordal graph.

Cordial labeling of hypertrees

A k-edge product cordial labelling is a variant of the well-known cordial labelling. In this paper, a balanced k-edge product cordial labelling is suggested and some sufficient conditions for balanced 3-edge product cordial graphs are proved. Moreover, a construction of graphs admitting a balanced 3-edge product cordial labelling is presented.

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate the 3-difference cordial labeling of wheel, helms, flower graph, sunflower graph, lotus inside a circle, closed helm, and double wheel.

For many parallel and distributed systems, automatic data redistribution improves its locality and increases system performance for various computer problems and applications. In general, an array can be distributed to multiple processing systems by using regular or irregular distributions. Some data distribution adopts BLOCK, CYCLIC, or BLOCK-CYCLIC to specify data array decomposition and distribution. On the other hand, irregular distributions specify a different-size data array distribution according to user-defined commands or procedures. In this work, we propose three bipartite graph problems, including the “maximum edge coloring problem”, the “maximum degree edge coloring problem”, and the “cost-sharing maximum edge coloring problem” to formulate these kinds of distribution problems. Next, we propose an approximation algorithm with a ratio bound of two for the maximum edge coloring problem when the input graph is biplanar. Moreover, we also prove that the “cost-sharing maximum edge coloring problem” is an NP-complete problem even when the input graph is biplanar.

We present in this paper an experimental study of six heuristics for a well-studied NP-complete graph problem: the vertex cover. These algorithms are adapted to process huge graphs. Indeed, executed on a current laptop computer, they offer reasonable CPU running times (between twenty seconds and eight hours) on graphs for which sizes are between 200 ·106 and 100 ·109 vertices and edges.We have run algorithms on specific graph families (we propose generators) and also on random power law graphs. Some of these heuristics can produce good solutions. We give here a comparison and an analysis of results obtained on several instances, in terms of quality of solutions and complexity, including running times.Keywordsimplementation of algorithmsexperimental analysishuge graphslow memoryvertex cover

Algorithms for graphs with small octopus