Hamiltonian Completion Number Research Articles

The formula for Turán number of spanning linear forests

Let F be a family of graphs. The Turán number ex(n;F) is defined to be the maximum number of edges in a graph of order n that is F-free. In 1959, Erdős and Gallai determined the Turán number of Mk+1 (a matching of size k+1) as follows: ex(n;Mk+1)=max2k+12,n2−n−k2.Since then, there has been a lot of research on Turán number of linear forests.A linear forest is a graph whose connected components are all paths or isolated vertices. Let Ln,k be the family of all linear forests of order n with k edges. In this paper, we prove that ex(n;Ln,k)=maxk2,n2−n−k−122+c,where c=0 if k is odd and c=1 otherwise. This determines the maximum number of edges in a non-Hamiltonian graph with given Hamiltonian completion number and also solves two open problems in Wang and Yang (2019) as special cases.Moreover, we show that our main theorem implies Erdős–Gallai Theorem and also gives a short new proof for it by the closure and counting techniques. Finally, we generalize our theorem to a conjecture which implies the famous Erdős Matching Conjecture.

Local search algorithms for finding the Hamiltonian completion number of line graphs

Given a graph G=(V,E), the Hamiltonian completion number of G, HCN(G), is the minimum number of edges to be added to G to make it Hamiltonian. This problem is known to be \(\mathcal{NP}\) -hard even when G is a line graph. In this paper, local search algorithms for finding HCN(G) when G is a line graph are proposed. The adopted approach is mainly based on finding a set of edge-disjoint trails that dominates all the edges of the root graph of G. Extensive computational experiments conducted on a wide set of instances allow to point out the behavior of the proposed algorithms with respect to both the quality of the solutions and the computation time.

Parallel algorithms for Hamiltonian problems on quasi-threshold graphs

In this paper we show structural and algorithmic properties on the class of quasi-threshold graphs, or QT-graphs for short, and prove necessary and sufficient conditions for a QT-graph to be Hamiltonian. Based on these properties and conditions, we construct an efficient parallel algorithm for finding a Hamiltonian cycle in a QT-graph; for an input graph on n vertices and m edges, our algorithm takes O( log n) time and requires O( n+ m) processors on the CREW PRAM model. In addition, we show that the problem of recognizing whether a QT-graph is a Hamiltonian graph and the problem of computing the Hamiltonian completion number of a nonHamiltonian QT-graph can also be solved in O( log n) time with O( n+ m) processors. Our algorithms rely on O( log n) -time parallel algorithms, which we develop here, for constructing tree representations of a QT-graph; we show that a QT-graph G has a unique tree representation, that is, a tree structure which meets the structural properties of G. We also present parallel algorithms for other optimization problems on QT-graphs which run in O( log n) time using a linear number of processors.

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

Given a graph G=( V, E), HCN( L( G)) is the minimum number of edges to be added to its line graph L( G) to make L( G) Hamiltonian. This problem is known to be NP-hard for general graphs, whereas a O(| V|) algorithm exists when G is a tree. In this paper a linear algorithm for finding HCN( L( G)) when G is a cactus is proposed.

Reduction Algorithms for Graphs of Small Treewidth

This paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth. S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese (J. Assoc. Comput. Mach.40, 1134–1164 (1993)) have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth, using a finite set of reduction rules. These algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first. We show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth, including all problems definable in monadic second order logic. We also show that a variant of these reduction algorithms can be used to solve (constructive) optimization problems in O(n) time. For example, optimization and construction variants of INDEPENDENT SET and HAMILTONIAN COMPLETION NUMBER can be solved in this way on graphs of small treewidth. Additionally, we show that the results of H. L. Bodlaender and T. Hagerup (SIAM J. Comput.27, 1725–1746 (1998)) can be applied to our reduction algorithms, which results in parallel reduction algorithms that use O(n) operations and O(log n log* n) time on an EREW PRAM, or O(log n) time on a CRCW PRAM.

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

A linear algorithm for the Hamiltonian completion number of the line graph of a tree

Given a graph G=(V,E), HCN(L(G)) is the minimum number of edges to be added to its line graph L(G) to make L(G) Hamiltonian. This problem is known to be NP-hard for general graphs, whereas an O(|V| 5) algorithm exists when G is a tree. In this paper a linear algorithm for finding HCN(L(G)) when G is a tree is proposed.

Powers of Hamiltonian paths in interval graphs

We give a simple proof that the obvious necessary conditions for a graph to contain the kth power of a Hamiltonian path are sufficient for the class of interval graphs. The proof is based on showing that a greedy algorithm tests for the existence of Hamiltonian path powers in interval graphs. We will also discuss covers by powers of paths and analogues of the Hamiltonian completion number. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 31–38, 1998

The total interval number of a tree and the Hamiltonian completion number of its line graph

In this paper we investigate the problem of computing two parameters of a graph: the Total Interval Number TIN( G), and the Hamiltonian Completion Number HCN( G). We show that in case of a triangle-free graph these two parameters (TIN( G) and HCN( L( G)), where L( G) is the line graph of G) are related by a simple formula. We use this relationship to find a polynomial algorithm which determines TIN( G), for a tree G. Then we describe a construction which gives an interval representation of a tree G with TIN( G) number of intervals.

A linear algorithm for the hamiltonian completion number of a tree

A linear algorithm for the hamiltonian completion number of a tree