Hybrid bases in graphs

International Journal of Circuit Theory and Applications, 1993

2020

The metric representation of a vertex v with respect to an ordered subset W = {w1, w2, · · · , wn} ⊆ V (G) is an ordered k−tuple defined by r(v|W ) = (d(v, w1), d(v, w2), . . . , d(v, wn)), where d(u, v) denotes the distance between the vertices u and v. A subset W ⊆ V (G) is a resolving set if all vertices of G have distinct representations with respect to W . A resolving set of the largest order whose no proper subset resolves all vertices of G is called the upper basis of G and the cardinality of the upper basis is called the upper dimension of G. A vertex v having at least one pendent edge incident on it is called a star vertex and the number of pendent edges incident on a vertex v is called the star degree of v. We determine the upper dimension of certain families of graphs and characterize the cases in which upper dimension equals the metric dimension. For instance, it is shown that metric dimension equals upper dimension for the graphs defined by the Cartesian product of Kn a...

Discrete Applied Mathematics, 2004

We investigate the class of graphs deÿned by the property that every induced subgraph has a vertex which is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set). In particular we give the list of minimal forbidden subgraphs for the subclass of graphs whose vertex-set can be emptied out by ÿrst recursively eliminating simplicial vertices and then recursively eliminating co-simplicial vertices.

Discussiones Mathematicae Graph Theory, 1998

In this paper, we introduce the notion of a variety of graphs closed under isomorphic images, subgraph identifications and induced subgraphs (induced connected subgraphs) firstly and next closed under isomorphic images, subgraph identifications, circuits and cliques. The structure of the corresponding lattices is investigated.

The IMA Volumes in Mathematics and its Applications, 2016

How can redundancy be eliminated from a network, and what can be said about the resulting substructures? In graph-theoretic terms, these substructures are spanning trees. The collection of all spanning trees of a graph has the structure of a matroid basis system; this observation connects trees to algebraic combinatorics and explains why many graph algorithms can be made computationally efficient. The number of spanning trees of a graph measures its complexity as a network, and there are classical and efficient linear-algebraic tools for calculating this number, as well as for enumerating trees more finely. The algebra of trees is key in studying dynamical systems on graphs, notably the abelian sandpile model (known in other forms as the chip-firing game or dollar game), whose possible states are encoded by a group of size equal to the complexity of the underlying graph.

1989

Discrete Mathematics, 2008

Let G = (V , E) be a graph. A set S ⊆ V is a dominating set of G if every vertex not in S is adjacent with some vertex in S. The domination number of G, denoted by (G), is the minimum cardinality of a dominating set of G. A set S ⊆ V is a paired-dominating set of G if S dominates V and S contains at least one perfect matching. The paired-domination number of G, denoted by p (G), is the minimum cardinality of a paired-dominating set of G. In this paper, we provide a constructive characterization of those trees for which the paired-domination number is twice the domination number.

Journal of The London Mathematical Society-second Series, 1971

In [1, 2] R. Rado proved several theorems on universal graphs. It is the purpose of this note to point out that the substance of these theorems, in fact strengthenings of them, can be obtained from general algebraic results of B. Jonsson or from related model theoretic results of M. Morley and R. Vaught. However, as we explain below, it seems to be the case that not all Rado's results can be obtained in this way, a fact which gives a little added interest to the comparison of results.

Journal of Combinatorial Theory, Series B, 1979

A matroidal family V is defined to be a collection of graphs such that, for any given graph G, the subgraphs of G isomorphic to a graph in V satisfy the matroid circuit-axioms. Here matroidal families closed under homeomorphism are considered. A theorem of Simks-Pereira shows that when only finite connected graphs are allowed as members of Q, two matroids arise: the cycle matroid and bicircular matroid. Here this theorem is generalized in two directions: the graphs are allowed to be infinite, and they are allowed to be disconnected. In the first case four structures result and in the second case two infinite families of rnatroids are obtained. The main theorem concerns the structures resulting when both restrictions are relaxed simultaneously. We will use standard graph theory terminology as far as possible, as found in [l], [2], or [13]. All graphs will be undirected and possibly infinite, and loops and multiple edges will be allowed. If G is a graph, E(G) denotes the set of edges of G and G\e denotes the graph obtained from G by deleting the edge e. A graph H is homeomorphic from G if it is isomorphic to a graph obtained from G by replacing each edge by a finite path and a graph K is homeomorphic to G if there exists some graph H such that G and K are both homeomorphic from H. The matroid theory terminology will follow [12]. One of the many ways to define a matroid on a finite set is by means of its collection % of circuits, which satisfies the following two axioms: (Cl) No member of V properly contains another.

2022

We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors, µ satisfies µ + < κ. Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property and for each λ, µ < κ there exists κ * < κ such that every tree of height µ with λ nodes has less than κ * branches.