Mihály Hjuter

Bio: Mihály Hjuter is an academic researcher. The author has contributed to research in topics: Cubic graph & Path graph. The author has an hindex of 1, co-authored 1 publications receiving 95 citations.

The number of maximal independent sets in triangle-free graphs

Abstract: In this paper, it is proved that every triangle-free graph on $n \geq 4$ vertices has at most $2^{n /2} $ or $5 \cdot 2^{( n - 5 )/2} $ independent sets maximal under inclusion, whether n is even or odd In each case, the extremal graph is unique If the graph is a forest of odd order, then the upper bound can be improved to $2^{( n - 1 )/2} $

Graph colorings with local constraints - a survey

Abstract: We survey the literature on those variants of the chromatic number problem where not only a proper coloring has to be found (i.e., adjacent vertices must not receive the same color) but some further local restrictions are imposed on the color assignment. Mostly, the list colorings and the precoloring extensions are considered. In one of the most general formulations, a graph G =( V, E), sets L(v) of admissible colors, and natural numbers cv for the vertices v ∈ V are given, and the question is whether there can be chosen a subset C(v) ⊆ L(v) of cardinality cv for each vertex in such a way that the sets

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

Abstract: We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterizations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m lO(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2nn O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732n) and exponential space. We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three. We extend the analysis to a number of related problems such as TSP and Chromatic Number.

Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques

Abstract: We give substantially improved exact exponential-time algorithms for a number of NP-hard problems These algorithms are obtained using a variety of techniques These techniques include: obtaining exact algorithms by enumerating maximal independent sets in a graph, obtaining exact algorithms from parameterized algorithms and a variant of the usual branch-and-bound technique which we call the "colored" branch-and-bound technique These techniques are simple in that they avoid detailed case analyses and yield algorithms that can be easily implemented We show the power of these techniques by applying them to several NP-hard problems and obtaining new improved upper bounds on the running time The specific problems that we tackle are: (1) the Odd Cycle Transversal problem in general undirected graphs, (2) the Feedback Vertex Set problem in directed graphs of maximum degree 4, (3) Feedback Arc Set problem in tournaments, (4) the 4-Hitting Set problem and (5) the Minimum Maximal Matching and the Edge Dominating Set problems The algorithms that we present for these problems are the best known and are a substantial improvement over previous best results For example, for the Minimum Maximal Matching we give an O*(14425n) algorithm improving the previous best result of O*(14422m) [35] For the Odd Cycle Transversal problem, we give an O*(162n) algorithm which improves the previous time bound of O*(17724n) [3]

On Independent Sets and Bicliques in Graphs

Abstract: Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3 n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642 n ), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.