Induced Subgraph Saturated Graphs

• Main Author: Craig Tennenhouse
• Format: Article
• Language: English
• Published: Georgia Southern University 2016-01-01
• Series: Theory and Applications of Graphs
• Subjects: graph saturation; induced subgraphs; paths; cycles; extremal graphs
• Online Access: https://digitalcommons.georgiasouthern.edu/tag/vol3/iss2/1

A graph $G$ is said to be \emph{$H$-saturated} if $G$ contains no subgraph isomorphic to $H$ but the addition of any edge between non-adjacent vertices in $G$ creates one. While induced subgraphs are often studied in the extremal case with regard to the removal of edges, we extend saturation to induced subgraphs. We say that $G$ is \emph{induced $H$-saturated} if $G$ contains no induced subgraph isomorphic to $H$ and the addition of any edge to $G$ results in an induced copy of $H$. We demonstrate constructively that there are non-trivial examples of saturated graphs for all cycles and an infinite family of paths and find a lower bound on the size of some induced path-saturated graphs.