Abstract.A Halin graph is a graph H = T ∪ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the end-vertices of T in the cyclic order determined by a plane embedding of T. In this paper, we define classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties.

We investigate here how far we can extend the notion of a Halin graph such that hamiltonicity is preserved. Let $H = T \cup C$ be a Halin graph, $T$ being a tree and $C$ the outer cycle. A $k$-Halin graph $G$ can be obtained from $H$ by adding edges while keeping planarity, joining vertices of $H - C$, such that $G - C$ has at most $k$ cycles. We prove that, in the class of cubic $3$-connected graphs, all $14$-Halin graphs are hamiltonian and all $7$-Halin graphs are $1$-edge hamiltonian. These results are best possible.

Interconnection systems in computer science and information technology are mainly represented by graphs. One such instance is of swapped network simulated by the optical transpose interconnection system (OTIS). Fault tolerance has become a vital feature of optoelectronic systems. Among multiple types of faults that may take place in an interconnection system, two significant kinds are either due to malfunctioning of a node (processor in case of O G ) or collapse of communication between nodes (failure of interprocessor transmission). To prevail over these faults, the unique recognition of every node is essential. In graph-theoretic interpretation, this leads to instigating the metric dimension β O G and fault-metric dimension β   ′ O G of the graph O G obtained from the interconnection system. This paper explores OTIS over base graph P m (path graph over m vertices) for resolvability and fault-tolerant resolvability. Furthermore, bounds for β O G and β ′ O G are also imparted over G = P m .

No abstract