Tutte Polynomials of Some Graphs

Given any graph G, there is a bivariate polynomial called Tutte polynomial which can be derived from G. We denote such polynomial by T(G; x; y). This thesis introduces the two techniques commonly used to compute T(G; x; y) along with several examples. Further, we determine T(G; x; y) for various classes of graphs such as cycles, trees, cacti, (2; 2; 1), which is a multi-bridge graph, and the well-known Peterson graph. We plot these surfaces, their contours and, for each such graph G, weevaluate their T(G; x; y) for some values (x; y) along a curve. We obtain important information about these graphs namely the number of spanning trees and number of spanning subgraphs. We also introduced some related polynomials such as thechromatic polynomial, the flow polynomial and the reliability polynomial

Algorithms for subgraph complementation to some classes of graphs

For a class $\mathcal{G}$ of graphs, the objective of \textsc{Subgraph Complementation to} $\mathcal{G}$ is to find whether there exists a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph in $\mathcal{G}$. We obtain a polynomial-time algorithm for the problem when $\mathcal{G}$ is the class of graphs with minimum degree at least $k$, for a constant $k$, answering an open problem by Fomin et al. (Algorithmica, 2020). When $\mathcal{G}$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices (for any constant $t\geq 3$) and diamond, we obtain a polynomial-time algorithm for the problem. This is in contrast with a result by Antony et al. (Algorithmica, 2022) that the problem is NP-complete and cannot be solved in subexponential-time (assuming the Exponential Time Hypothesis) when $\mathcal{G}$ is the class of graphs without any induced copies of the star graph on $t+1$ vertices, for every constant $t\geq 5$

Gamma graphs of some special classes of trees

A set S⊂V is a dominating set of a graph G=(V,E) if every vertex v∈V which does not belong to S has a neighbour in S. The domination number γ(G) of the graph G is the minimum cardinality of a dominating set in G. A dominating set S is a γ-set in G if |S|=γ(G). "Some graphs have exponentially many γ-sets, hence it is worth to ask a question if a γ-set can be obtained by some transformations from another γ-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second section gamma graphs γ.T of trees with diameter not greater than five will be presented. It will be shown that hypercubes Qk are among γ.T graphs. In the third section γ.T graphs of certain trees with three pendant vertices will be analysed. Additionally, some observations on the diameter of gamma graphs will be presented, in response to an open question, published by Fricke et al., if diam(T(γ))=O(n)