A star-factor of a graph G is a spanning subgraph of G such that each component is a star. An edge-weighting of G is a function w : E(G) −→ N, where N is the set of positive integers. Let Ω be the family of all graphs G such that every star-factor of G has the same weight under some fixed edge-weighting w. The open problem of characterizing the class Ω, posed by Hartnell and Rall, is motivated ...

Let G be a multipartite multigraph without loops. Then G is said to be internally fair if its edges are shared as evenly as possible among all pairs of its partite sets. An internally fair factorization of G is an edge-decomposition of G into internally fair regular spanning subgraphs. A holey factor of G is a regular subgraph spanning all vertices but one partite set. An internally fair holey ...

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if for every vertex v ∈ V (G), the subgraph induced by the neighborhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, w...

Abstract A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 11]. We define a simple 2-factor as a 2-factor without degenerate cycles. In general, simple k-factors are defined as k-regular spanning subgrap...

A star-factor of a graph G is a spanning subgraph of G such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph G is called star-uniform if all star-factors of G have the same number of components. To characterize star-uniform graphs was an open problem posed by Hartnell and Rall, which is motivated by the minimum cost spanning tree an...

Topology Control in an Indoor Wireless Sensor Network using Realistic Propagation Models Gagan Goel Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2014 This thesis considers the problem of designing an energy-efficient communication backbone for a wireless sensor network deployed in an indoor environment. We propose and implement a Top...

Let G = ({1, . . . , n}, E, w) be an undirected graph with nonnegative edge weights w and let aij be the nonnegative requirement between vertices i and j. For any spanning subgraph H of G, the weight of H is the total weight of its edges and the routing cost of H is ∑ i<j aijdH(i, j), where dH(i, j) is the distance between i and j in H. In this paper, we investigated two special cases of the pr...

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting...

Define a graph to be a Kotzig graph if it is m-regular and has an m-edge colouring in which each pair of colours form a Hamiltonian cycle. We show that every cubic graph with spanning subgraph consisting of a subdivision of a Kotzig graph together with even cycles has a cycle double cover, in fact a 6-CDC. We prove this for two other families of graphs similar to Kotzig graphs as well. In parti...

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-...