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Graph Classes.md

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Graph Classes

#graphTheory

Trees

Trees are defined by the following equivalent definitions:

  • A Tree is a connected graph without cycles.

  • It is a connected graph on n vertices with n - 1 edges.

  • A graph is a tree iff there is a unique simple path between any pair of its vertices.

  • A path graph is always a tree.

  • A spanning tree of a graph $G$, is a subgraph of $G$ which is a tree containing all vertices of $G$.

  • A minimum spanning tree of a weighted graph $G$ is a spanning tree of the smallest weight. The problem of finding a minimum spanning tree has implications for optimizing network connectivity.

Bipartite Graphs

  • A graph $G$ is Bipartite if its vertices can be partitioned into two disjoint sets $L$ and $R$ such that:
    • Every edge of $G$ connects a vertex $L$ to a vertex in $R$.
    • No edge connects two vertices from the same set.
  • $L$ and $R$ are called the parts of $G$. ![[Pasted image 20230724194906.png]]
  • A graph is bipartite, $iff$ it has no cycles of odd length.
  • A cycle graph with even vertices is a bipartite graph. Not true if number of vertices is odd.
  • Trees are bipartite as well.
  • A Complete Bipartite Graph $K_{L, R}$ is a bipartite graph in which all vertices of $L$ are connected to all vertices of $R$ and vice versa.

Matchings

  • A matching in a graph is a set of edges without common vertices.

  • A maximal matching is a matching which cannot be extended to a larger matching.

  • A maximum matching is a matching of the largest size.

  • In bipartite graphs, matchings can be used to find mapping solutions from one part to the other.

Hall's Theorem

  • **In a bipartite graph $G = (L \cup R, E)$, there is a matching which covers all vertices from L, $iff$ for every subset of vertices $S \subseteq L$, $$|S| \leq |N(S)|$$
  • Here, the Neighborhood $N(S)$ of $S$, a subset of vertices $S \subseteq V$ of graph $G = (V, E)$, is defined as the set of all vertices connected to at least one vertex in $S$.

Planar Graphs

  • A graph is said to be planar if it can be drawn in the plane such that its edges do not meet (cross) except at their end points.
  • Geographical maps are always equivalent to planar graphs. ![[Pasted image 20230724202237.png]]

Euler's Formula

Let $G$ be a connected planar graph, then $$v - e + f = 2$$ Where, $v$ is the number of vertices, $e$ is the number of edges, and $f$ is the number of faces in the drawing of $G$. ![[Pasted image 20230724202807.png]]