Fast minimal triangulation algorithm using minimum degree criterion

AbstractWe propose an algorithm for minimal triangulation which, using simple and efficient strategy, subdivides the input graph in different, almost non-overlapping, subgraphs. Using the technique of matrix multiplication for saturating the minimal separators, we show that the partition of the graph can be computed in time O(nα) where nα is the time required by the binary matrix multiplication. After saturating the minimal separators, the same procedure is recursively applied on each subgraphs. We also present a variant of the algorithm in which the minimum degree criterion is used. In this way, we obtain an algorithm that uses minimum degree criterion and at the same time produces a minimal triangulation, thus shedding new light on the effectiveness of the minimum degree heuristics

Equivalence between Hypergraph Convexities

Let G be a connected graph on V. A subset X of V is all-paths convex (or ap -convex) if X contains each vertex on every path joining two vertices in X and is monophonically convex (or m-convex) if X contains each vertex on every chordless path joining two vertices in X. First of all, we prove that ap -convexity and m-convexity coincide in G if and only if G is a tree. Next, in order to generalize this result to a connected hypergraph H, in addition to the hypergraph versions of ap -convexity and m-convexity, we consider canonical convexity (or c-convexity) and simple-path convexity (or sp -convexity) for which it is well known that m-convexity is finer than both c-convexity and sp -convexity and sp -convexity is finer than ap -convexity. After proving sp -convexity is coarser than c-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of γ-acyclic hypergraphs

A polynomial quantum computing algorithm for solving the dualization problem

Given two prime monotone boolean functions $f:\{0,1\}^n \to \{0,1\}$ and $g:\{0,1\}^n \to \{0,1\}$ the dualization problem consists in determining if $g$ is the dual of $f$, that is if $f(x_1, \dots, x_n)= \overline{g}(\overline{x_1}, \dots \overline{x_n})$ for all $(x_1, \dots x_n) \in \{0,1\}^n$. Associated to the dualization problem there is the corresponding decision problem: given two monotone prime boolean functions $f$ and $g$ is $g$ the dual of $f$? In this paper we present a quantum computing algorithm that solves the decision version of the dualization problem in polynomial time

Quantum invariants for the graph isomorphism problem

Graph Isomorphism is such an important problem in computer science, that it has been widely studied over the last decades. It is well known that it belongs to NP class, but is not NP-complete. It is thought to be of comparable difficulty to integer factorisation. The best known proved algorithm to solve this problem in general, was proposed by László Babai and Eugene Luks in 1983. Recently, there has been some research in the topic by using quantum computing, that also leads the present piece of research. In fact, we present a quantum computing algorithm that defines an invariant over Graph Isomorphism characterisation. This quantum algorithm is able to distinguish more non-isomorphic graphs than most of the known invariants so far. The proof of correctness and some hints illustrating the extent and reason of the improvement are also included in this paper

Fully dynamic algorithm for chordal graphs with O(1) query-time and O(n2) update-time

We propose dynamic algorithms and data structures for chordal graphs supporting the following operation: determine if an edge can be added or removed from the graph while preserving the chordality in O(1) time. We show that the complexity of the algorithms for updating the data structures when an edge is actually inserted or deleted is O(n2) where n is the number of vertices of the graph

On the geodetic iteration number of the contour of a graph

Let G be a graph and S be a subset of vertices of G. With I[S] we denote the set of all vertices on some geodesic (shortest path) between two vertices of S. A contour vertex of a graph is one whose eccentricity is at least as big as all its neighbors' eccentricities. Let C be the set of contour vertices of a graph. We provide the first example of a graph where I[I[C]] do not coincide with the vertex set of the graph

Polynomial time algorithm for computing a minimum geodetic set in outerplanar graphs

Given a graph G and a pair of vertices the interval is the set of all vertices that are in some shortest path between u and v. Given a subset X of vertices of G, the interval of X, is the union of the intervals for all pairs of vertices in X and we say that X is geodetic if its interval do coincide with the set of vertices in the graph. A minimum geodetic set is a minimum cardinality geodetic set of G. The problem of computing a minimum geodetic set is known to be NP-Hard for general graphs but is known to be polynomially solvable for maximal outerplanar graphs. In this paper we show a polynomial time algorithm for finding a minimum geodetic set in general outerplanar graphs