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 .

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In this write-up, we explore the family of two-bridge knots for fault-tolerance. We findout fault-tolerant unknotting numbers for certain subfamilies of two-bridge knots. Besidesthis, we work out on special subfamilies C(a, a1, a2, a3, . . . ,±2,−ak,−ak−1, . . . ,−a2,−a1)and C(2n, 3) of two-bridge knots with unknotting number 1 and n respectively and found avarient of fault-tolerant unknotting number called restricetd fault-tolerant unknotting number.Lastly, this paper is concluded by explicating biological disposition of fault-tolerantunknotting number in terms of enzyme action on DNA.

In this study, we introduce the locating-dominating value and the location-domination polynomial of graphs and location-domination polynomials of some families of graphs were identified. Locatingdominating set of graph G is defined as the dominating set which locates all the vertices of G. And, location-domination number G is the minimum cardinality of a locating-dominating set in G.