2021

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Browsing 2021 by Author "Muhammad Ahsan"

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    Computing edge metric dimension of planar graphs
    (UMT Lahore, 2021-12-15) Muhammad Ahsan
    Let K = (V (K), E(K)) be a connected graph and x, y ∈ V (K), d(x, y) = min{ length of x − y path } and for e = ab ∈ E(K), d(x, e) = min{d(x, a), d(x, b)}. A vertex x distinguishes two edges e1 and e2 if d(e1, x) ≠ d(e2, x). For an edge e of K and a subset WE = {w1, w2, . . . , wk} of its vertices, the representation of e with respect to WE , denoted by r(e | WE ), is the k-tuple (d(e, w1), d(e, w2), . . . , d(e, wk)). If distinct edges of K have distinct representation with respect to WE , then WE is called an edge metric generator (EMG) for K. An EMG of minimum cardinality is an edge metric basis (EMB) for K, and its cardinality is called edge metric dimension (EMD) of K, denoted by edim(K). In this thesis, the constant EMD in the form of exact and upper bound for the graphs the cycle with chord graph, kayak paddle graph, tadpole graph, the cartesian product of cycle with path graph, the necklace graph, circulant graphs, the prism related graph, toeplitz networks are computed. It is also studied that the flower graph and some prism related graph have unbounded EMD. Further, the study of fault-tolerant edge metric dimension (FEMD) is initiated in this work. An EMG ́WE of K is called fault-tolerant edge metric generator (FEMG) of K if ́WE \ {v} is also an EMG of graph K for every v ∈ ́WE . An FEMG of minimum cardinality is a fault-tolerant edge metric basis (FEMB) for graph K, and its cardinality is called FEMD of K. The FEMD of the path, cycle, complete graph, cycle with chord graph, tadpole graph, and kayak paddle graph was also computed.