Let $$G$$ be a simple undirected graph. Let $${T_D}$$ be a depth first search tree of $$G.$$ Let $${T_B}$$ be a breadth first search tree of $$G.$$ Consider the following statements.

$$(I)$$ No edge of $$G$$ is a cross edge with respect to $${T_D}.$$ ($$A$$ cross edge in $$G$$ is between
$$\,\,\,\,\,\,\,\,$$ two nodes neither of which is an ancestor of the other in $${T_D}.$$)
$$(II)$$ For every edge $$(u,v)$$ of $$G,$$ if $$u$$ is at depth $$i$$ and $$v$$ is at depth $$j$$ in $${T_B}$$, then
$$\,\,\,\,\,\,\,\,\,\,\,$$ $$\left| {i - j} \right| = 1.$$

Which of the statements above must necessarily be true?

Let $$G$$ be a graph with $$100!$$ vertices, with each vertex labelled by a distinct permutation of the numbers $$1,2, … , 100.$$ There is an edge between vertices $$u$$ and $$v$$ if and only if the label of $$u$$ can be obtained by swapping two adjacent numbers in the label of $$v.$$ Let $$𝑦$$ denote the degree of a vertex in $$G,$$ and $$𝑧$$ denote the number of connected components in $$G.$$ Then, $$𝑦 + 10𝑧 =$$ _____.

In an adjacency list representation of an undirected simple graph $$G = (V,E),$$ each edge $$(u, v)$$ has two adjacency list entries: $$[v]$$ in the adjacency list of $$u,$$ and $$[u]$$ in the adjacency list of $$v.$$ These are called twins of each other. A twin pointer is a pointer from an adjacency list entry to its twin. If $$|E| = m$$ and $$|V| = n,$$ and the memory size is not a constraint, what is the time complexity of the most efficient algorithm to set the twin pointer in each entry in each adjacency list?

Let $$G$$ be a connected undirected graph of $$100$$ vertices and $$300$$ edges. The weight of a minimum spanning tree of $$G$$ is $$500.$$ When the weight of each edge of $$G$$ is increased by five, the weight of a minimum spanning tree becomes ________.