Proof Compression and NP Versus PSPACE II: Addendum

In [3] we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier's cut-free sequent calculus for minimal logic (HSC) [5] with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction (ND) [6]. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea (due to the second author) is to omit full minimal logic and compress only \naive" normal tree-like ND refutations of the existence of Hamiltonian cycles in given non-Hamiltonian graphs, since the Hamiltonian graph problem in NP-complete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE [3].

Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

The H-force sets of the graphs satisfying the condition of Ore’s theorem

Let G be a Hamiltonian graph. A nonempty vertex set X\subseteq V(G) is called a Hamiltonian cycle enforcing set (in short, an H-force set) of G if every X-cycle of G (i.e., a cycle of G containing all vertices of X) is a Hamiltonian cycle. For the graph G, h(G) (called the H-force number of G) is the smallest cardinality of an H-force set of G. Ore’s theorem states that an n-vertex graph G is Hamiltonian if d(u)+d(v)\ge n for every pair of nonadjacent vertices u,v of G. In this article, we study the H-force sets of the graphs satisfying the condition of Ore’s theorem, show that the H-force number of these graphs is possibly n, or n-2 , or \frac{n}{2} and give a classification of these graphs due to the H-force number.

Locally Hamiltonian Graphs and Minimal Size of Maximal Graphs on a Surface

We prove that every locally Hamiltonian graph with $n$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a graph $G$ on some 2-dimensional surface $\Sigma$ (not necessarily compact) has at least $3n-6$ edges with equality if and only if $G$ also triangulates the sphere. If, in addition, $G$ is simple, then for each vertex $v$, the cyclic ordering of the edges around $v$ on $\Sigma$ is the same as the clockwise or anti-clockwise orientation around $v$ on the sphere. If $G$ contains no complete graph on 4 vertices, then the face-boundaries are the same in the two embeddings.

CHARACTER GRAPHS WITH NONBIPARTITE HAMILTONIAN COMPLEMENT

For a finite group $G$, let $\unicode[STIX]{x1D6E5}(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we obtain a necessary and sufficient condition which guarantees that the complement of the character graph $\unicode[STIX]{x1D6E5}(G)$ of a finite group $G$ is a nonbipartite Hamiltonian graph.

A Novel Rule-Based Method for Individualized Spot Welding Sequence Optimization With Respect to Geometrical Quality

Spot welding is the predominant joining process for the sheet metal assemblies. The assemblies, during this process, are mainly bent and deformed. These deformations, along with the single part variations, are the primary sources of the aesthetic and functional geometrical problems in an assembly. The sequence of welding has a considerable effect on the geometrical variation of the final assembly. Finding the optimal weld sequence for the geometrical quality can be categorized as a combinatorial Hamiltonian graph search problem. Exhaustive search to find the optimum, using the finite element method simulations in the computer-aided tolerancing tools, is a time-consuming and thereby infeasible task. Applying the genetic algorithm to this problem can considerably reduce the search time, but finding the global optimum is not guaranteed, and still, a large number of sequences need to be evaluated. The effectiveness of these types of algorithms is dependent on the quality of the initial solutions. Previous studies have attempted to solve this problem by random initiation of the population in the genetic algorithm. In this paper, a rule-based approach for initiating the genetic algorithm for spot weld sequencing is introduced. The optimization approach is applied to three automotive sheet metal assemblies for evaluation. The results show that the proposed method improves the computation time and effectiveness of the genetic algorithm.