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On Convex Sets and Convexity Numbers of Some Graphs

Let G be a graph without isolated vertices. An edge in G is said to cover the vertices with which it is incident. A subset U of E(G) is an edge cover of G if for each vertex v \in V(G) there is an edge in U which covers v. The edge covering number of G is given by e_c(G)=\text{min}\left \{ \left | U \right |:U\, \textup{is an edge cover} \right \}. This study seeks to determine the edge covering numbers of graphs which result from some unary or binary operations and other simple graphs.

For any graph without isolated vertices, the edge set of a graph is an edge cover of the graph. Hence, the size of a graph is an upper bound for its edge covering number. The edge covering number of a graph is equal to its size if and only if it is a star.

If a graph has a spanning path, then its edge covering number is equal to the least integer greater than or equal to half of its order. Some upper bounds for the sum, the cartesian product, the composition, and the corona of two graphs, and graphs which result from some unary operations are also obtained in this study.

Cited as: Artes, Rosalio Jr. G.. On Edge Cover of Graphs. Master’s Thesis, MSU-IIT. April 2004.
Adviser: Sergio R. Canoy, Jr., Ph.D.

Category: Mathematics
Course/Degree: Master of Science in Mathematics
School/Institution/University: School of Graduate Studies, Mindanao State University-Iligan Institute of Technology
School Address: Tibanga, Iligan City, Philippines
Published in: 2004