To find any random spanning tree of a graph a simple DFS will obviously suffice. Show that a spanning tree of the complete graph K 4 is either a depth-first spanning tree or a breadth-first spanning tree. Assuming the graph is connected, the edges that we traversed during the DFS will form the spanning tree edge set. For example in the graph above, vertices 4 and 8 couldn't possibly have a back-edge connecting them because neither of them is an ancestor of the other. Viewed 135 times 0. Are the spanning forests created by DFS and by BFS solutions to some graph optimization problems? Active 5 years, 10 months ago. (b) Find a spanning tree of the complete graph K 5 which is neither a depth-first nor a breadth-first spanning tree. Ask Question Asked 5 years, 10 months ago. Depth-first search (DFS) is an algorithm for searching a graph or tree data structure. De nition 1.0.3. We use Stack data structure with maximum size of total number of vertices in the graph to implement DFS traversal. In 1970, Klaus Wagner (  p.50) posed a problem of characterizing con-nected graphs in which any two spanning trees are isomorphic. So DFS of a tree is relatively easier. In 1971, Bohdan Zelinka  published a solution obtained by considering invariants of a tree. On undirected graph G, a DFS tree starting at vertex s visits all vertices on the connected component of s. The discovery edges (the edges in the DFS tree) form a spanning tree over the connected component of s. On a directed graph, a DFS tree starting at vertex s visits all vertices that are reachable from s. The algorithm starts at the root (top) node of a tree and goes as far as it can down a given branch (path), then backtracks until it finds an unexplored path, and then explores it. Spanning tree DFS algorithm doesn't create a tree. The algorithm does this until the entire graph has been explored. We want to show that we will get a spanning... Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The following figure shows a minimum spanning tree on an edge-weighted graph: Similarly, a maximum spanning tree has the largest weight among all spanning trees. digraph is strongly connected, so the dfs-tree produced will not necessarily be a spanning tree. The function dfs-nextArc selects and returns as its value the frontier arc whose tree-endpoint has the largest dfnumber. Same can be done using a BFS too. This is why DFS tree is so useful. Why? DFS Traversal of a Graph vs Tree. The back-edges of the graph all connect a vertex with its descendant in the spanning tree. The following figure shows a maximum spanning tree on an edge-weighted graph: 3. 2. When a depth- rst search is executed on a digraph, In this case, each time we visit a new node for the first time, we add the parent edge to the spanning tree set. A minimum spanning tree is a spanning tree whose weight is the smallest among all possible spanning trees. Unlike graph, tree does not contain cycle and always connected. Spanning Tree is a graph without loops. Def 2.4. In graph, there might be cycles and dis-connectivity. Posted by 2 days ago In depth first search (preorder or postorder) and breadth first search, spanning forests of the original graph are created. We can simply begin from a node, then traverse its adjacent (or children) without caring about cycles. DFS (Depth First Search) BFS (Breadth First Search) DFS (Depth First Search) DFS traversal of a graph produces a spanning tree as final result. Proposition 2.2. A spanning tree of a graph Gis a spanning subgraph of G that is a tree. 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