Enumerating Hamiltonian Cycles in A 2-connected Regular Graph Using Planar Cycle Bases

Planar fundamental cycle basis belong to a 2-connected simple graph is used for enumerating Hamiltonian cycles contained in the graph. This is because a fun- damental cycle basis is easily constructed. Planar basis is chosen since it has a weighted induced graph whose values are limited to 1. Hence making it is possible to be used in the Hamiltonian cycle enumeration procedures efficiently. In this paper a Hamiltonian cycle enumeration scheme is obtained through two stages. Firstly, i cycles out of m bases cycles are determined using an appropriate con- structed constraint. Secondly, to search all Hamiltonian cycles which are formed by the combination of i basis cycles obtained in the first stage efficiently. This ef- ficiency is achieved through the generation of a class of objects consisting of Ill-bit binary strings which is a representation of i cycle combinations between m cycle basis cycle

Algorithmic approaches to circuit enumeration problems and applications

June 1982Also issued as an M.S. thesis, Dept. of Aeronautics and Astronautics, 1982Includes bibliographical references (p. 129-132)A review of methods of enumerating elementary cycles and circuits is presented. For the directed planar graph, a geometric view of circuit generation is introduced making use of the properties of dual graphs. Given the set of elementary cycles or circuits, a particular algorithm is recommended to generate all simple circuits. A simple example accompanies each of the methods discussed. Some methods of reducing the size of the graph but maintaining all circuits are introduced. Worst-case bounds on computational time and space are also given. The problem of enumerating elementary circuits whose cost is less than a certain fixed cost is solved by modifying an existing algorithm. The cost of a circuit is the sum of the cost of the arcs forming the circuit where arc costs are not restricted to be positive. Applications of circuits with particular properties are suggested

Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph

In this paper, we address the problem of enumerating all induced subtrees in an input k-degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to k, and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced subgraph versions have not been studied well. One of few example is for chordless paths and cycles. Our motivation is to reduce the time complexity close to O(1) for each solution. This type of optimal algorithms are proposed many subgraph classes such as trees, and spanning trees. Induced subtrees are fundamental object thus it should be studied deeply and there possibly exist some efficient algorithms. Our algorithm utilizes nice properties of k-degeneracy to state an effective amortized analysis. As a result, the time complexity is reduced to O(k) time per induced subtree. The problem is solved in constant time for each in planar graphs, as a corollary

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability