What is n-connected graph?

A graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater.

This means a graph G is said to be k-connected if there does not exist a set of k-1 vertices whose removal disconnects the graph

Problem Statement Given is a weighted undirected connected graph G with N vertices and M edges, which may contain self-loops and multi-edges. The vertices are labeled as Vertex 1, Vertex 2, …, Vertex N. The edges are labeled as Edge 1, Edge 2, …, Edge M. Edge i connects Vertex a i ​ and Vertex b i ​ and has a weight of c i ​ . Here, for every pair of integers (i,j) such that 1≤i<j≤M, c i ​  =c j ​ holds. Process the Q queries explained below. The i-th query gives a triple of integers (u i ​ ,v i ​ ,w i ​ ). Here, for every integer j such that 1≤j≤M, w i ​  =c j ​ holds. Let e i ​ be an undirected edge that connects Vertex u i ​ and Vertex v i ​ and has a weight of w i ​ . Consider the graph G i ​ obtained by adding e i ​ to G. It can be proved that the minimum spanning tree T i ​ of G i ​ is uniquely determined. Does T i ​ contain e i ​ ? Print the answer as Yes or No. Note that the queries do not change T. In other words, even though Query i considers the graph obtained by adding e i ​ to G, the G in other queries does not have e i ​ . What is minimum spanning tree? The spanning tree of G is a tree with all of the vertices in G and some of the edges in G. The minimum spanning tree of G is the tree with the minimum total weight of edges among the spanning trees of G. Constraints 2≤N≤2×10 5 N−1≤M≤2×10 5 1≤a i ​ ≤N (1≤i≤M) 1≤b i ​ ≤N (1≤i≤M) 1≤c i ​ ≤10 9 (1≤i≤M) c i ​  =c j ​ (1≤i<j≤M) The graph G is connected. 1≤Q≤2×10 5 1≤u i ​ ≤N (1≤i≤Q) 1≤v i ​ ≤N (1≤i≤Q) 1≤w i ​ ≤10 9 (1≤i≤Q) w i ​  =c j ​ (1≤i≤Q,1≤j≤M) All values in input are integers.c++code
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03-23
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