An undirected graph is acyclic (i.e., a forest) if a DFS yields no back edges. Since back edges are those edges ( u , v ) connecting a vertex u to an ancestor v in a depth-first tree, so no back edges means there are only tree edges, so
there is no cycle
.
What is an undirected graph with no cycles is called?
ii) An undirected graph which contains no cycles is called
a forest
.
Can a spanning tree have cycles?
A spanning tree is a subset of Graph G, which has all the vertices covered with minimum possible number of edges. Hence,
a spanning tree does not have cycles
and it cannot be disconnected..
Can a connected graph have cycles?
For a graph with n > 3 vertices, note that by Theorem 1,
a 2-regular graph with n vertices has n edges and by Corollary 3.1 such a graph must contain a cycle
. Since the graph is 2- regular, none of the vertices in this cycle can be contained in any other edges, so the cycle is a connected component.
How does BFS detect cycle in undirected graph?
We do a BFS traversal of the given graph. For every visited vertex ‘v’, if there is an adjacent ‘u’ such that u is already visited and u is not a parent of v, then there is a cycle in the graph. If we don’t find such an adjacent for any vertex, we say that there is no cycle.
What is undirected graph in data structure?
An undirected graph is
a set of nodes and a set of links between the nodes
. Each node is called a vertex, each link is called an edge, and each edge connects two vertices. The order of the two connected vertices is unimportant. An undirected graph is a finite set of vertices together with a finite set of edges.
Can a graph have multiple minimum spanning tree?
A graph can have more than one MST in the case where both trees have the same overall weight but different paths to complete the tree
.
Why graph traversal is difficult than tree traversal?
Graph data structures structures are a bit more complex than trees because
they can have loops, circuits and self-loops see the (1, 2, 3) loops in traversals
. Graphs therefore tend to be more connected and complex than trees. The bi-directional nature of some graphs also adds to the complexity.
Does every graph have a spanning tree?
Every finite connected graph has a spanning tree
. However, for infinite connected graphs, the existence of spanning trees is equivalent to the axiom of choice. An infinite graph is connected if each pair of its vertices forms the pair of endpoints of a finite path.
How many edges does an undirected graph have?
The maximum number of edges in an undirected graph is
n(n-1)/2
and obviously in a directed graph there are twice as many. Show activity on this post. If the graph is not a multi graph then it is clearly n * (n – 1), as each node can at most have edges to every other node.
What is undirected graph give its properties?
Undirected graphs
have edges that do not have a direction
. The edges indicate a two-way relationship, in that each edge can be traversed in both directions. This figure shows a simple undirected graph with three nodes and three edges. Directed graphs have edges with direction.
Is a connected undirected graph without any cycle?
A graph is connected if there is a path from every vertex to every other vertex. A graph that is not connected consists of a set of connected components, which are maximal connected subgraphs.
An acyclic graph is a graph with no cycles
. A tree is an acyclic connected graph.
Can a simple graph have cycles?
A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex. In other words
a simple graph is a graph without loops
and multiple edges.
Does BFS work for undirected graphs?
The only reason to use a BFS would be if you know your (undirected) graph is going to have long paths and small path cover
(in other words, deep and narrow). In that case, BFS would require proportionally less memory for its queue than DFS’ stack (both still linear of course).
Which of the following can be used for cycle detection in undirected graph?
Depth First Traversal
can be used to detect a cycle in a Graph. DFS for a connected graph produces a tree.
How do I find cycles with BFS?
Steps involved in detecting cycle in a directed graph using BFS. Step-1: Compute in-degree (number of incoming edges) for each of the vertex present in the graph and initialize the count of visited nodes as 0. Step-3: Remove a vertex from the queue (Dequeue operation) and then. Increment count of visited nodes by 1.
Is undirected graph connected?
An undirected graph is graph, i.e., a set of objects (called vertices or nodes) that are
connected together
, where all the edges are bidirectional. An undirected graph is sometimes called an undirected network. In contrast, a graph where the edges point in a direction is called a directed graph.
How does an undirected graph differ from a directed graph?
The main difference between directed and undirected graph is that
a directed graph contains an ordered pair of vertices whereas an undirected graph contains an unordered pair of vertices
. A graph is a nonlinear data structure that represents a pictorial structure of a set of objects that are connected by links.
What is an undirected unweighted graph?
An unweighted undirected graph is
defined by vertices and edges
. The vertices are also often called the nodes in the graph. Vertices may be connected by edges. A common definition of a graph is via its adjacency matrix.
Can a given undirected weighted graph have more than one MST?
The answer would, probably be no ; multiple STs are possible (depends on the graph), but,
if each edge has a unique weight , only one MST will be possible
.
Can two minimum spanning trees for the same graph have different edge weights?
That would be impossible
, though, because we know that T2 is an MST. Therefore, we know that there is some other edge in the cycle whose weight is equal to w*. If any of those edges are not in T1, then choose any one and remove it.
Why the minimum spanning tree of a graph may not be unique?
If the edge weights are all positive, it suffices to define the MST as the subgraph with minimal total weight that connects all the vertices. The edge weights are all different.
If edges can have equal weights
, the minimum spanning tree may not be unique.
How are cycles detected in a typical graph traversal problem Mcq?
Solution:
A back edge means a cycle in graph
. So if there is a cycle, all DFS traversals would contain at least one back edge.
What will be the sum of degrees of each vertices for undirected graph G if it has n vertices and e edges?
Explanation: Since the given graph is undirected,
every edge contributes as 2 to sum of degrees
.
Are complete graphs regular?
Ans: A graph is said to be regular if all the vertices are of same degree.
Yes a complete graph is always a regular graph
.
