CHARACTERIZATION OF SPAN OF BASE B-INDUCED
1-UNIFORM DCSL GRAPHS
K.A. Germina1, Gency Joseph2, L. Benedict Michael Raj3
1Department of Mathematics
Central University of Kerala
Kasargod, Kerala, INDIA
2 Department of Mathematics
St. Joseph's College, Trichy
Tamilnadu, INDIA

Abstract

A distance compatible set labeling (dcsl) of a connected graph $G$ is an injective set assignment $f : V(G) \rightarrow 2^{X},$ $X$ being a non empty ground set, such that the corresponding induced function $f^{\oplus} :E(G) \rightarrow 2^{X}\setminus \{\phi\}$ given by $f^{\oplus}(uv)= f(u)\oplus f(v)$ satisfies $\vert f^{\oplus}(uv)\vert = k_{(u,v)}^{f}d_{G}(u,v) $ for every pair of distinct vertices $u, v \in V(G),$ where $d_{G}(u,v)$ denotes the path distance between $u$ and $v$ and $k_{(u,v)}^{f}$ is a constant, not necessarily an integer, depending on the pair of vertices $u,v$ chosen. A dcsl $f$ of $G$ is $k$-uniform if all the constants of proportionality with respect to $f$ are equal to $k,$ and if $G$ admits such a dcsl then $G$ is called a $k$-uniform dcsl graph. Let $\mathcal{F}$ be a family of subsets of a set $X.$ A tight path between two distinct sets $P$ and $Q$ in ${\mathcal{F}}$ is a sequence $P_0=P, P_1, P_2 \dots P_n = Q$ in ${\mathcal{F}}$ such that $d(P,Q)= \mid P \bigtriangleup Q \mid = n$ and $d(P_i, P_{i+1}) = 1$ for $0 \leq i \leq n-1.$ The family $\mathcal F$ is well-graded family, if there is a tight path between any two of its distinct sets. In this paper we characterize problem of determining those $\mathcal F$-induced graph $G_\mathcal F$ in which the base $\mathcal B$-induced graph is $1$-uniform dcsl.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 1
Year: 2019

DOI: 10.12732/ijam.v32i1.5

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