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A Fan-Type Heavy Pair Of Subgraphs For Pancyclicity Of 2-Connected Graphs

Wojciech Wideł (2016)

Discussiones Mathematicae Graph Theory

Let G be a graph on n vertices and let H be a given graph. We say that G is pancyclic, if it contains cycles of all lengths from 3 up to n, and that it is H-f1-heavy, if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies [...] min⁡dG(u),dG(v)≥n+12 min { d G ( u ) , d G ( v ) } ≥ n + 1 2 . In this paper we prove that every 2-connected K1,3, P5-f1-heavy graph is pancyclic. This result completes the answer to the problem of finding f1-heavy pairs of subgraphs implying pancyclicity...

A few remarks on the history of MST-problem

Jaroslav Nešetřil (1997)

Archivum Mathematicum

On the background of Borůvka’s pioneering work we present a survey of the development related to the Minimum Spanning Tree Problem. We also complement the historical paper Graham-Hell [GH] by a few remarks and provide an update of the extensive literature devoted to this problem.

A Fiedler-like theory for the perturbed Laplacian

Israel Rocha, Vilmar Trevisan (2016)

Czechoslovak Mathematical Journal

The perturbed Laplacian matrix of a graph G is defined as L D = D - A , where D is any diagonal matrix and A is a weighted adjacency matrix of G . We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore, we use...

A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

Yury Metelsky, Kseniya Schemeleva, Frank Werner (2017)

Discussiones Mathematicae Graph Theory

We characterize the class [...] L32 L 3 2 of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 L 3 2 in the class of threshold graphs, where n is the number of vertices of a tested graph.

A formula for all minors of the adjacency matrix and an application

R. B. Bapat, A. K. Lal, S. Pati (2014)

Special Matrices

We supply a combinatorial description of any minor of the adjacency matrix of a graph. This description is then used to give a formula for the determinant and inverse of the adjacency matrix, A(G), of a graph G, whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.

A Gallai-type equality for the total domination number of a graph

Sanming Zhou (2004)

Discussiones Mathematicae Graph Theory

We prove the following Gallai-type equality γₜ(G) + εₜ(G) = p for any graph G with no isolated vertex, where p is the number of vertices of G, γₜ(G) is the total domination number of G, and εₜ(G) is the maximum integer s such that there exists a spanning forest F with s the number of pendant edges of F minus the number of star components of F.

A game of composing binary relations

P. GoralČík, Z. Hedrlín, V. Koubek, J. Ryšunková (1982)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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