 Spanning tree

A spanning tree of a connected graph $G$ is a subgraph that is a tree and contains every vertex of $G$. Its edges must come from $G$. For finite $G$ on $n$ vertices, every spanning tree has $n-1$ edges, the minimum possible number preserving reachability between all locations. This makes a spanning tree a compact certificate that the original graph is connected. A minimum spanning tree additionally minimizes a specified total edge weight; merely selecting a spanning tree supplies no such optimization. ^[ref]

Let $G$ be our seven-vertex tree with edge $57$ added. It has seven edges and a graph cycle. Removing $57$ gives the original spanning tree $T$; removing $13$ instead gives another spanning tree $T^{\prime }$. Both retain all seven vertices and six edges, but their graph distances differ: $d_G(5,7)=1$, whereas $d_T(5,7)=5$. In $T^{\prime }$, $d_{T^{\prime }}(5,7)=1$ but $d_{T^{\prime }}(1,3)=5$, along $1,2,5,7,6,3$, compared with $d_G(1,3)=1$. Preserving reachability permits longer journeys; breadth-first search can preserve graph distances from one chosen starting vertex.

To construct a spanning tree, repeatedly delete any edge on a graph cycle. Its endpoints remain joined through the rest of that graph cycle, so the deletion preserves connectedness. Finiteness ensures that this stops at a connected acyclic graph. Conversely, adding an omitted edge to a spanning tree creates its unique fundamental cycle; deleting any other edge on that graph cycle produces another spanning tree. The displayed exchange replaces $13$ by $57$. ^[ref]

A **spanning tree** is a concept from graph theory and is particularly important in the field of computer science, networking, and related disciplines. Here’s a breakdown of the concept: 1. **Definition**: A spanning tree of a graph is a subgraph that includes all the vertices of the original graph and is connected, without any cycles. This means it is a tree structure that spans all the vertices in the graph.