Consider a graph G = (V, E) with nonnegative integer function c : V → N. Find an augmenting path method to compute a subgraph H = (V, F) (F ⊆ E) with maximum number of edges such that for every v ∈ V , deg(v) ≤ c(v)

Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.

Square graph. The square of a directed graph G = (V,E) is the graph G^2 =(V,E^2)suchthat(u,w)∈E^2 if and only iff or some v∈V both(u,v)∈E and (v, w) ∈ E; that is, there is a path of exactly two edges. (i) Suppose that G is represented by an adjacency list. Describe an efficient algorithm for computing G^2 from G. (ii) Suppose that G is represented by an adjacency matrix. Describe an efficient algorithm for computing G^2 from G.

Let G = (V,E) be a simple graph which is weighted, undirected, and connected. Sup- pose G contains a unique edge having the largest weight. Let emax be this edge. Suppose removing emax in G does not disconnect G. Prove that any minimum spanning tree of G must not contain the edge emax. Hint: Prove by contradiction. (Consider on the contrary that the MST contains emax. Think in this way: Can we replace this edge by some other edge to reduce the total weight?)

Given a graph G = (V;E) an almost independent set I  V is a generalization of an independent set whereeach vertex u 2 I is connected to at most one other vertex in I but no more. In other words for all u; v 2 Iif (u; v) 2 E then (u; z) =2 E for all z 2 I (equvalently (v; z) =2 E). Note that every independent set is alsoalmost independent. Prove that the problem of  nding whether there exists an almost independent set ofsize k for some k, is NP-complete.

We have a connected graph G=(V,E), and a specific vertex u∈V. Suppose we compute a depth-first search tree rooted at u, and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u, and obtain the same tree T.   Prove that G=T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.)

Let G (V, E) be a digraph in which every vertex is a source, or a sink, or both a sink and a source. (a) Prove that G has neither self-loops nor anti-parallel edges.

Prove that there exists a maximum flow f in G, such that the set of edges carrying positive flowis acyclic. In other words, f has the property that for every directed cycle in G made up of edgese1, e2, . . . , ek, at least one of the values f (e1), f (e2), . . . , f (ek) is equal to zero.

Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.

Let G = (V, E) denote an weighted undirected graph, in which every edge has unit weight, and let T = (V, E') denote the minimum spanning tree of G. Prove formally that the graph G need not have a unique tree T