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Title: Computing Edge Metric Dimension of Planar Graphs
Authors: Ahsan, Muhammad
Lahore, University
Keywords: Mathematics
Physical Sciences
Issue Date: 2021
Publisher: University of Management & Technology, Lahore
Abstract: Let K = (V (K);E(K)) be a connected graph and x; y 2 V (K), d(x; y) = minf length of x 􀀀 y path g and for e = ab 2 E(K), d(x; e) = minfd(x; a); d(x; b)g. A vertex x distinguishes two edges e1 and e2 if d(e1; x) 6= d(e2; x). For an edge e of K and a subset WE = fw1;w2; : : : ;wkg of its vertices, the representation of e with respect toWE, denoted by r(e j 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 n fvg is also an EMG of graph K for every v 2 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.
Gov't Doc #: 26681
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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