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Minimal trees and monophonic convexity

Jose Cáceres, Ortrud R. Oellermann, M. L. Puertas (2012)

Discussiones Mathematicae Graph Theory

Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is...

Minimal vertex degree sum of a 3-path in plane maps

O.V. Borodin (1997)

Discussiones Mathematicae Graph Theory

Let wₖ be the minimum degree sum of a path on k vertices in a graph. We prove for normal plane maps that: (1) if w₂ = 6, then w₃ may be arbitrarily big, (2) if w₂ < 6, then either w₃ ≤ 18 or there is a ≤ 15-vertex adjacent to two 3-vertices, and (3) if w₂ < 7, then w₃ ≤ 17.

Minimality of toric arrangements

Giacomo d'Antonio, Emanuele Delucchi (2015)

Journal of the European Mathematical Society

We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

Kamal Lochan Patra, Binod Kumar Sahoo (2013)

Czechoslovak Mathematical Journal

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g ( n , g being fixed), which graph minimizes the Laplacian spectral radius? Let U n , g be the lollipop graph obtained by appending a pendent vertex of a path on n - g ( n > ...

Minimum convex-cost tension problems on series-parallel graphs

Bruno Bachelet, Philippe Mahey (2003)

RAIRO - Operations Research - Recherche Opérationnelle

We present briefly some results we obtained with known methods to solve minimum cost tension problems, comparing their performance on non-specific graphs and on series-parallel graphs. These graphs are shown to be of interest to approximate many tension problems, like synchronization in hypermedia documents. We propose a new aggregation method to solve the minimum convex piecewise linear cost tension problem on series-parallel graphs in O ( m 3 ) operations.

Minimum convex-cost tension problems on series-parallel graphs

Bruno Bachelet, Philippe Mahey (2010)

RAIRO - Operations Research

We present briefly some results we obtained with known methods to solve minimum cost tension problems, comparing their performance on non-specific graphs and on series-parallel graphs. These graphs are shown to be of interest to approximate many tension problems, like synchronization in hypermedia documents. We propose a new aggregation method to solve the minimum convex piecewise linear cost tension problem on series-parallel graphs in O(m3) operations.

Minimum degree, leaf number and traceability

Simon Mukwembi (2013)

Czechoslovak Mathematical Journal

Let G be a finite connected graph with minimum degree δ . The leaf number L ( G ) of G is defined as the maximum number of leaf vertices contained in a spanning tree of G . We prove that if δ ≥ 1 2 ( L ( G ) + 1 ) , then G is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if δ ≥ 1 2 ( L ( G ) + 1 ) , then G contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin. 15 (2008),...

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