An algorithm for finding a similar subgraph of all Hamiltonian cycles

This paper discusses an algorithm to find a similar subgraph called findSimSubG algorithm. A similar subgraph is a subgraph with a maximum number of edges, contains no isolated vertex and is contained in every Hamiltonian cycle of a Hamiltonian Graph. The algorithm runs only on Hamiltonian graphs with at least two Hamiltonian cycles. The algorithm works by examining whether the initial subgraph of the first Hamiltonian cycle is a subgraph of comparison graphs. If the initial subgraph is not in comparison graphs, the algorithm will remove edges and vertices of the initial subgraph that are not in comparison graphs. There are two main processes in the algorithm, changing Hamiltonian cycle into a cycle graph and removing edges and vertices of the initial subgraph that are not in comparison graphs. The findSimSubG algorithm can find the similar subgraph without using backtracking method. The similar subgraph cannot be found on certain graphs, such as an n -antiprism graph, complete bipartite graph, complete graph, 2 n -crossed prism graph, n -crown graph, n -möbius ladder, prism graph, and wheel graph. The complexity of this algorithm is O ( m | V |), where m is the number of Hamiltonian cycles and | V | is the number of vertices of a Hamiltonian graph.