A path of length two is denoted by P2. If a graph G does not contain P2 as induced subgraph, then: 1- G must be a clique (i.e., a complete graph). 2- Every vertex of G must of degree one. 3- Every connected component of G must be a clique. 4- Every connected component of G must consist of at most two vertices.

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Minimum Spanning Tree

A spanning tree is a sub-graph of a connected, undirected graph. It has all the vertices of a graph with the minimum possible number of edges. If a vertex is skipped, then it will not be a spanning tree. The edges of a spanning tree may or may not have weights.