4 research outputs found
Characterization of removable elements with respect to having k disjoint bases in a matroid
AbstractThe well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with k edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with k disjoint bases. For any graph G that may not have k-edge-disjoint spanning trees, the problem of determining what edges should be added to G so that the resulting graph has k edge-disjoint spanning trees has been studied by Haas (2002) [11] and Liu et al. (2009) [17], among others. This paper aims to determine, for a matroid M that has k disjoint bases, the set Ek(M) of elements in M such that for any eâEk(M), Mâe also has k disjoint bases. Using the matroid strength defined by Catlin et al. (1992) [4], we present a characterization of Ek(M) in terms of the strength of M. Consequently, this yields a characterization of edge sets Ek(G) in a graph G with at least k edge-disjoint spanning trees such that âeâEk(G), Gâe also has k edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set Ek(M) in a matroid M, or the edge subset Ek(G) for a connected graph G
Densities in graphs and matroids
Certain graphs can be described by the distribution of the edges in its subgraphs.
For example, a cycle C is a graph that satisfies |E(H)|
|V (H)| < |E(C)|
|V (C)| = 1 for all non-trivial
subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)|
|V (H)|â1 †|E(T)|
|V (T)|â1 = 1
for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such
that |E(H)|
|V (H)| †|E(G)|
|V (G)| and a 1-balanced graph is a graph such that |E(H)|
|V (H)|â1 †|E(G)|
|V (G)|â1
for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G
that satisfy the property |E(H)| †k|V (H)| â l for all non-trivial subgraphs H of G
play important roles in defining rigid structures.
This dissertation is a formal study of a class of density functions that extends the
above mentioned ideas. For a rational number r †1, a graph G is said to be r-balanced
if and only if for each non-trivial subgraph H of G, we have |E(H)|
|V (H)|âr †|E(G)|
|V (G)|âr . For
r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined
and the existence of these graphs is discussed. We also define a class of vulnerability
measures on graphs similar to the edge-connectivity of graphs and show how it is
related to r-balanced graphs. All these definitions are matroidal and the definitions
of r-balanced matroids naturally extend the definitions of r-balanced graphs.
The vulnerability measures in graphs that we define are ranked and are lesser
than the edge-connectivity. Due to the relationship of the r-balanced graphs with
the vulnerability measures defined in the dissertation, identifying r-balanced graphs
and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced
matroids and their weak forms are discussed. For r â {0, 1}, we give a method to
construct big r-balanced graphs from small r-balanced graphs. This construction is a
generalization of the construction of Cartesian product of two graphs. We present an
algorithmic solution of the problem of transforming any given graph into a 1-balanced
graph on the same number of vertices and edges as the given graph. This result is
extended to a density function defined on the power set of any set E via a pair of
matroid rank functions defined on the power set of E. Many interesting results may
be derived in the future by choosing suitable pairs of matroid rank functions and
applying the above result
Cycles and Bases of Graphs and Matroids
The objective of this dissertation is to investigate the properties of cycles and bases in matroids and in graphs. In [62], Tutte defined the circuit graph of a matroid and proved that a matroid is connected if and only if its circuit graph is connected. Motivated by Tutte\u27s result, we introduce the 2nd order circuit graph of a matroid, and prove that for any connected matroid M other than U1,1, the second order circuit graph of M has diameter at most 2 if and only if M does not have a restricted minor isomorphic to U2,6.;Another research conducted in this dissertation is related to the eulerian subgraph problem in graph theory. A graph G is eulerian if G is connected without vertices of odd degrees, and G is supereulerian if G has a spanning eulerian subgraph. In [3], Boesch, Suffey and Tindel raised a problem to determine when a graph is supereulerian, and they remarked that such a problem would be a difficult one. In [55], Pulleyblank confirmed the remark by showing that the problem to determine if a graph is supereulerian, even within planar graphs, is NP-complete. Catlin in [8] introduced a reduction method based on the theory of collapsible graphs to search for spanning eulerian subgraphs in a given graph G. In this dissertation, we introduce the supereulerian width of a graph G, which generalizes the concept of supereulerian graphs, and extends the supereulerian problem to the supereulerian width problem in graphs. Further, we also generalize the concept of collapsible graphs to s-collapsible graphs and develop the reduction method based on the theory of s-collapsible graphs. Our studies extend the collapsible graph theory of Catlin. These are applied to show for any integer n \u3e 2, the complete graph Kn is (n - 3)- collapsible, and so the supereulerian width of Kn is n - 2. We also prove a best possible degree condition for a simple graph to have supereulerian width at least 3.;The number of edge-disjoint spanning trees plays an important role in the design of networks, as it is considered as a measure of the strength of the network. As disjoint spanning trees are disjoint bases in graphic matroids, it is important to study the properties related to the number of disjoint bases in matroids. In this dissertation, we develop a decomposition theory based on the density function of a matroid, and prove a decomposition theorem that partitions the ground set of a matroid M into subsets based on their densities. As applications of the decomposition theorem, we investigate problems related to the properties of disjoint bases in a matroid. We showed that for a given integer k \u3e 0, any matroid M can be embedded into a matroid M\u27 with the same rank (that is, r(M) = r( M\u27)) such that M\u27 has k disjoint bases. Further we determine the minimum value of |E( M\u27)| -- |E(M)| in terms of invariants of M. For a matroid M with at least k disjoint bases, we characterize the set of elements in M such that removing any one of them would still result in a matroid with at least k disjoint bases