From 26950f685bf29ab4ec5aea3ccbc3933e37fbf130 Mon Sep 17 00:00:00 2001 From: dimbo4ka Date: Tue, 11 Feb 2025 14:02:41 +0300 Subject: [PATCH] The definition of graph isomorphism was added. Information about graph K2 walk was added --- cheat5.tex | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/cheat5.tex b/cheat5.tex index 8e10452d..c179c82c 100644 --- a/cheat5.tex +++ b/cheat5.tex @@ -98,6 +98,13 @@ \section{Graph Theory Cheatsheet% \item $A^{(k)} = \Set{\Set{x_1,\dotsc,x_k} \given x_1 \neq \dotsb \neq x_k \in A} = \Set{ S \given S \subseteq A, \card{S} = k }$ is the set of $k$-sized subsets of~$A$. \end{terms} + \item Two graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ are called \textbf{isomorphic} if there exists a bijection $f: V_1 \to V_2$ such that for all $u, v \in V_1$, the edge relation is preserved: + \[ + \Set{u, v} \in E_1 \iff \Set{f(u), f(v)} \in E_2. + \] + This means that the graphs are structurally identical up to vertex renaming. + Notation: $G_1 \cong G_2$. + \item Simple \textbf{directed}\Href{https://en.wikipedia.org/wiki/Directed_graph} graphs have $E \subseteq V^{2}$, \ie each edge $e_i \in E$ from vertex $u$ to~$v$ is denoted by an ordered pair~$\Pair{u,v} \in V^{2}$. Such \emph{directed edges} are also called \emph{arcs} or \emph{arrows}. @@ -364,7 +371,7 @@ \section{Graph Theory Cheatsheet% Walk & $+$ & $+$ & Closed walk \\ Trail & $+$ & $-$ & Circuit \\ Path & $-$ & $-$ & Cycle \\ - & $-$ & $+$ & (\emph{impossible}) \\ + & $-$ & $+$ & (\emph{possible for $K_2$}) \\ \bottomrule \end{NiceTabular} \end{wrapfigure}