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Computing the Metric Dimension of a Graph from Primary Subgraphs

Dorota Kuziak, Juan A. Rodríguez-Velázquez, Ismael G. Yero (2017)

Discussiones Mathematicae Graph Theory

Let G be a connected graph. Given an ordered set W = {w1, . . . , wk} ⊆ V (G) and a vertex u ∈ V (G), the representation of u with respect to W is the ordered k-tuple (d(u, w1), d(u, w2), . . . , d(u, wk)), where d(u, wi) denotes the distance between u and wi. The set W is a metric generator for G if every two different vertices of G have distinct representations. A minimum cardinality metric generator is called a metric basis of G and its cardinality is called the metric dimension of G. It is well...

Conditions for β-perfectness

Judith Keijsper, Meike Tewes (2002)

Discussiones Mathematicae Graph Theory

A β-perfect graph is a simple graph G such that χ(G') = β(G') for every induced subgraph G' of G, where χ(G') is the chromatic number of G', and β(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs,...

Conditions nécessaires d’existence des ( k , r , s , ) -plans

G. Heuzé (1972)

Mathématiques et Sciences Humaines

Les ( k , r , s ) -plans (définis ci-dessous) ont été introduits dans [1]. Leur étude englobe celle des plans affines et projectifs finis, des familles de carrés latins deux à deux orthogonaux, de certains plans équilibrés et partiellement équilibrés 2 . La question de leur existence est très mal connue, celle de leur unicité n’a pratiquement pas été abordée. Nous nous proposons de montrer le théorème suivant : pour qu’il existe un ( k , r , s ) -plan il est nécessaire que : k ( k - 1 ) ( r - 1 ) s , r ( k - 1 ) ( r - 1 ) s , k r ( k - 1 ) ( r - 1 ) s ( k + r - s - 1 ) soient entiers.

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