The main focus of this thesis are hypercubes. In the first part, we introduce hypercubes, which form an interesting class of graphs that has practical uses in networks and distributed computing. Because of their varied applications, the thesis describes the graph-theory problems related to hypercubes such as searching for detour spanners, minimizing their maximal degree and finding multiple edge- disjoint spanners. It also overviews current results on selected hypercube problems and proposes a solution using a genetic algorithm. The genetic algorithm is designed, implemented and its performance is evaluated. The conclusion is that applying a genetic algorithm to some hypercube problems is a viable, but not the most effective method.

Given a complete graph K_n = (V,E) with non-negative edge costs c ∈ R^E, the problem multi-2EC_cost is that of finding a 2-edge connected spanning multi-subgraph of K_n with minimum cost. It is believed that there are no efficient ways to solve the problem exactly, as it is NP-hard. Methods such as approximation algorithms, which rely on lower bounds like the linear programming relaxation multi-2EC^LP of multi-2EC , thus become vital cost cost to obtain solutions guaranteed to be close to the optimal in a fast manner. In this thesis, we focus on the integrality gap αmulti-2EC of multi-2EC^LP , which is a measure of the quality of multi-2EC^LP as a lower bound. Although we currently only know cost that 6/5 ≤ αmulti-2EC_cost ≤ 3 , the integrality gap for multi-2EC_cost has been conjectured to be 6/5. We explore the idea of using the structure of solutions for αmulti-2EC_cost and the concept of convex combination to obtain improved bounds for αmulti-2EC_cost. We focus our efforts on a family J of half-integer solutions that appear to give the largest integrality gap for multi-2EC_cost. We successfully show that the conjecture αmulti-2EC_cost = 6/5 is true for any cost functions optimized by some x∗ ∈ J. We also study the related problem 2EC_size, which consists of finding the minimum size 2-edge connected spanning subgraph of a 2-edge connected graph. The problem is NP-hard even at its simplest, when restricted to cubic 3-edge connected graphs. We study that case in the hope of finding a more general method, and we show that every 3-edge connected cubic graph G = (V ′, E′), with n = |V ′| allows a 2EC_size solution for G of size at most 7n/6 This improves upon Boyd, Iwata and Takazawa’s guarantee of 6n/5 and extend Takazawa’s 7n/6 guarantee for bipartite cubic 3-edge connected graphs to all cubic 3-edge connected graphs.

A graph is said to be 2-edge-connected if it remains connected after the deletion of any single edge. Given an unweighted bridgeless graph G with n vertices, the minimum size 2-edge-connected spanning subgraph problem (2EC) is that of finding a 2-edge-connected spanning subgraph of G with the minimum number of edges. This problem has important applications in the design of survivable networks. However, because the problem is NP-hard, it is unlikely that efficient methods exist for solving it. Thus efficient methods that find solutions that are provably close to optimal are sought. In this thesis, an approximation algorithm is presented for 2EC on bridgeless cubic graphs which guarantees to be within 5/4 of the optimal solution value, improving on the previous best proven approximation guarantee of 5/4+ε for this problem. We also focus on the linear programming (LP) relaxation of 2EC, which provides important lower bounds for 2EC in useful solution techniques like branch and bound. The “goodness” of this lower bound is measured by the integrality gap of the LP relaxation for 2EC, denoted by α2EC. Through a computational study, we find the exact value of α2EC for graphs with small n. Moreover, a significant improvement is found for the lower bound on the value of α2EC for bridgeless subcubic graphs, which improves the known best lower bound on α2EC from 9/8 to 8/7.

A graph is said to be 2-edge-connected if it remains connected after the deletion of any single edge. Given an unweighted bridgeless graph G with n vertices, the minimum size 2-edge-connected spanning subgraph problem (2EC) is that of finding a 2-edge-connected spanning subgraph of G with the minimum number of edges. This problem has important applications in the design of survivable networks. However, because the problem is NP-hard, it is unlikely that efficient methods exist for solving it. Thus efficient methods that find solutions that are provably close to optimal are sought. In this thesis, an approximation algorithm is presented for 2EC on bridgeless cubic graphs which guarantees to be within 5/4 of the optimal solution value, improving on the previous best proven approximation guarantee of 5/4+ε for this problem. We also focus on the linear programming (LP) relaxation of 2EC, which provides important lower bounds for 2EC in useful solution techniques like branch and bound. The “goodness” of this lower bound is measured by the integrality gap of the LP relaxation for 2EC, denoted by α2EC. Through a computational study, we find the exact value of α2EC for graphs with small n. Moreover, a significant improvement is found for the lower bound on the value of α2EC for bridgeless subcubic graphs, which improves the known best lower bound on α2EC from 9/8 to 8/7.

In structural graph theory, connectivity is an important notation with a lot of applications. Tutte, in 1961, showed that a simple graph is 3-connected if and only if it can be generated from a wheel graph by repeatedly adding edges between nonadjacent vertices and applying vertex splitting. In 1971, Halin constructed a class of edge-minimal 3-connected planar graphs, which are a generalization of wheel graphs and later were named “Halin graphs” by Lovasz and Plummer. A Halin graph is obtained from a plane embedding of a tree with no stems having degree 2 by adding a cycle through its leaves in the natural order determined according to the embedding. Since Halin graphs were introduced, many useful properties, such as Hamiltonian, hamiltonian-connected and pancyclic, have been discovered. Hence, it will reveal many properties of a graph if we know the graph contains a spanning Halin subgraph. But unfortunately, until now, there is no positive result showing under which conditions a graph contains a spanning Halin subgraph. In this thesis, we characterize all forbidden pairs implying graphs containing spanning Halin subgraphs. Consequently, we provide a complete proof conjecture of Chen et al. Our proofs are based on Chudnovsky and Seymour’s decomposition theorem of claw-free graphs, which were published recently in a series of papers.