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\title[Outer-connected Edge Domination in Graphs]{Outer-connected Edge Domination in Graphs}
\author [Maquilan, C.J. - 2016-00168]{\textbf{Christle Jude L. Maquilan (BS Mathematics)}}
\institute[Western Mindanao State University] {\emph{Adviser: }\textbf{Mohammad Nur S. Paspasan, Ph.D.}\\[1em]
Department of Mathematics and Statistics\\College of Science and Mathematics\\Western Mindanao State University\\[1em]
\includegraphics[scale=0.09]{./Figures/WMSU CSM DMS.png}}
\date[April 21, 2022]{\footnotesize Math 197 (Special Problem) - Final Defense - \textbf{April 21, 2022}}
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\section{Introduction}
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\section{Working Definitions}
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\begin{definition}\label{d19} A set $M \subseteq E(G)$ is an \emph{edge dominating set of $G$} if every $u \in E(G) \backslash M$ is adjacent to some $v \in M$. The \emph{edge domination number of $G$}, denoted by $\gamma_{e}(G)$, is the minimum cardinality of an edge dominating set of $G$. Any edge dominating set of $G$ with cardinality $\gamma_{e}(G)$ is referred to as a \emph{$\gamma_{e}$-set of $G$}.
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\normalfont{The sets $M_{1}=\{a, c, f\}, M_{2}=\{d, h\}$, and $M_{3}=\{a, e, g, h\}$ are edge dominating sets of $G$ in Figure 1.5. Moreover, $M_{2}=\{d, h\}$ is a minimum edge dominating set of $G$. Thus, $\gamma_{e}(G)=\left|M_{2}\right|=2$.}
\begin{figure}[ht]
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\includegraphics[width=0.35\textwidth]{./Figures/EDOM.png}
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\caption{A graph $G$ with $\gamma_{e}(G)=2$.\label{edom}}
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\section{Results}
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\begin{remark}\label{rem: 1}
A set $S$ is an outer-connected edge dominating set of a graph $G$ if $S$ is an edge dominating set such that $H_{E(G) \backslash \mathrm{S}}$ does not have component isomorphic to $K_{2}$ or $S=E(G)$.
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\indent To see this, consider graphs $G_{1}=P_{3}, G_{2}=P_{4}$, and $G_{3}=C_{8}$ in Figure \ref{3.1}.
Then, $\gamma_{oce}(P_{3})=2, \gamma_{oce}(P_{4})=3$, and $\gamma_{o c e}(C_{8})=4$.
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\begin{frame}{Results (Cont'n)}
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\caption{Graphs with $\gamma_{oce}(P_{3})=2, \gamma_{oce}(P_{4})=3$, and $\gamma_{oce}(C_{8})=4$.\label{3.1}}
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\section{Recommendations}
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The following problems are suggested for further study:\\
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\nocite{akhbari2013outer,arumugam2009connected,berge1962theory}
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\frametitle{List of References}
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\Huge\emph{Thank You So Much!}
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