### Making Holes in the Second Symmetric Product of a Cyclicly Connected Graph

#### Abstract

A \textit{continuum} is a connected compact metric space. The \textit{second symmetric product} of a continuum $ X $, $ \mathcal{F}_2(X) $, is the hyperspace of all nonempty subsets of $ X $ having at most two elements. An element $ A $ of $ \mathcal{F}_2(X) $ is said to \textit{make a hole with respect to multicoherence degree} in $\mathcal{F}_2(X) $ if the multicoherence degree of $ \mathcal{F}_2(X) - \{A\} $ is greater than the multicoherence degree of $ \mathcal{F}_2(X) $. In this paper, we characterize those elements $A \in \mathcal{F}_2(X) $ such that $A$ makes a hole with respect to multicoherence degree in $ \mathcal{F}_2(X) $ when $ X $ is a cyclicly connected graph.

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Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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