The basis number of some special non-planar graphs

Discrete Mathematics, 1998

A basis of the cycle space C(G) of a graph G is h-fold if each edge of G occurs in at most h cycles of the basis. The basis number b(G) of G is the least integer h such that C(G) has an h-fold basis. MacLane showed that a graph G is planar if and only if b(G) ≤ 2. Schmeichel

The basis number b(G) of a graph G is defined to be the least integer d such that G has a d-fold basis for its cycle space. In this paper we: give an upper bound of the basis number of the direct product of trees; classify the trees with respect to the basis number of the direct product of trees and paths of order greater than or equal to 5; give an upper bound of the basis number of the direct product of bipartite graphs; and investigate the basis number of the direct product of a bipartite graph and a cycle.

Mathematical and Computer Modelling, 1991

It is shown that the number of labeled spanning trees of a connected graph is equal to the determinant of the intersection matrix for an integral basis of its cycles. The one-skeletons of a convex polyhedron and its dual have the same number of spanning trees. An integral cycle basis is constructed for graphs possessing rotational symmetry. Certain sequences of a priori algebraic numbers are seen to be computable using integers only.

Computer Science Review, 2009

Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and apriori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

Missouri Journal of Mathematical Sciences

The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. MacLane [13] proved that a graph G is planar if and only if the basis number of G is less than or equal to 2. Ali [3] proved that the basis number of the strong product of a path and a star is less than or equal to 4. In this work, (1) We give an appropriate decomposition of trees. (2) We give an upper bound of the basis number of a cycle and a bipartite graph. (3) We give an upper bound of the basis number of a path and a bipartite graph. This is a generalization of Ali's result [3]. 1. Introduction. Unless otherwise specified, all graphs considered here are connected, finite, undirected, and simple. We start by introducing the definitions of the following basic product graphs. Let G and H be two graphs. (1) The direct product G * = G ∧ H has vertex-set V (G) × V (H) and edge-set E(G *) = {(u 1 , u 2)(v 1 , v 2) | u 1 v 1 ∈ E(G) and u 2 v 2 ∈ E(H)}. (2) The cartesian product G * = G × H has vertex-set V (G *) = V (G) × V (H) and edge-set E(G *) = {(u 1 , u 2)(v 1 , v 2) | u 1 = v 1 and u 2 v 2 ∈ E(H) or u 2 = v 2 and u 1 v 1 ∈ E(G)}. (3) The strong product G * = G ⊗ H has vertex-set V (G *) = V (G) × V (H) and edge set E(G *) = {(u 1 , u 2)(v 1 , v 2) | u 1 = v 1 and u 2 v 2 ∈ E(H) or u 2 = v 2 and u 1 v 1 ∈ E(G) or u 1 v 1 ∈ E(G) and u 2 v 2 ∈ E(H)}. (4) The semi-strong product G * = G • H has vertex set V (G *) = V (G) × V (H) and edge set E(G *) = {(u 1 , u 2)(v 1 , v 2) | u 1 v 1 ∈ E(G) and u 2 v 2 ∈ E(H) or u 1 = v 1 and u 2 v 2 ∈ E(H)}. (5) The lexicographic product G * = G[H] has vertex set V (G *) = V (G) × V (H) and edge set E(G) = {(u 1 , u 2)(v 1 , v 2) | u 1 v 1 ∈ E(G) or u 1 = v 1 and u 2 v 2 ∈ E(H)}.

Journal of Graph Theory, 1991

showed that the planar graphs with no prism minor are the graphs obtainable by 2-sums from bonds, cycles, wheels and K,\e's. We give a new characterization of these graphs in terms of an optimization problem defined on the cycle bases of a graph.

Mathematical Problems in Engineering

Let G be a simple connected graph. Suppose Δ = Δ 1 , Δ 2 , … , Δ l an l -partition of V G . A partition representation of a vertex α w . r . t Δ is the l − vector d α , Δ 1 , d α , Δ 2 , … , d α , Δ l , denoted by r α | Δ . Any partition Δ is referred as resolving partition if ∀ α i ≠ α j ∈ V G such that r α i | Δ ≠ r α j | Δ . The smallest integer l is referred as the partition dimension pd G of G if the l -partition Δ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

Acta Mathematica Hungarica, 2004

he basis number, b(G), of a graph G is defined to be the least integer k such that G has a k-fold basis for its cycle space. In this paper we investigate the basis number of the composition of theta graphs with stars and wheels.

Journal of Combinatorial Theory, Series B, 1982

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...