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Assignment 9
Graphs
- Find the sum of the degrees of the vertices of each graph and verify that it equals twice the number of edges in the graph.
Ans:
deg(a) = deg(i) = 3 deg(b) = deg(h) = 2 deg(c) = deg(g) = 4 deg(d) = 0 deg(e) = 6 2 ݁ൌ 2 ݃݁݀ ൌ 12 ሺݒሻ ൌ 24 ௩∈
deg(a) = deg(b) = 4 deg(c) = deg(d) = deg(e) = 6
2 ݁ൌ 2 ݃݁݀ ൌ 13 ሺݒሻ ൌ 26 ௩∈
- Determine the number of vertices and edges and find the in-degree and out-degree of each vertex for the given directed multigraph.
Ans:
ି݃݁݀ ሺ ܽሻ ൌ 6, ି݃݁݀ ሺ ܾሻ ൌ 4, ି݃݁݀ ሺ ܿሻ ൌ 2, ି݃݁݀ ݀ሺ ሻ ൌ 1,
݃݁݀ ା^ ሺ ܽሻ ൌ 1, ݃݁݀ ା^ ሺ ܾሻ ൌ 2, ݃݁݀ ା^ ሺ ܿሻ ൌ 5, ݃݁݀ ା^ ݀ሺ ሻ ൌ 5.
௩∈
௩∈
- Determine whether each of the following graphs is bipartite
Ans: Not bipartite graph Yes; V 1 { a, b, d, e } and V 2 { c, f }
- Use an adjacency list to represent the given graph.
Ans: Adjacency List for a simple Graph Vertex Adjacent vertices a b c d
a, b, c, d d a, b b, c, d
Adjacency List for a simple Graph Vertex Adjacent vertices a b c d e
b, d a, c, d, e b, c a, e c, e
- Draw the graph represented by each of the given adjacency matrix.
Ans:
a. b. Similarly as part a.
- Use an incidence matrix to represent each of the following graphs
a. b.
Ans: a.
݁ଵ ݁ଶ ݁ଷ ݁ସ ݁ହ ݁ (^) ଼݁݁ ݁ଽ^ ଵ 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0
b. Similarly as part a.
- Determine whether the given pair of graphs is isomorphic.
Ans:
Yes; with f(u 1 ) = v 1 , f(u 2 ) = v 4 , f(u 3 ) = v 2 , f(u 4 ) = v 5 , and f(u 5 ) = v 3.
ܷൌ ۏ
ێ ێ ێ
ۍ^011001 11
(^0 1 0 1 ) 1 1
1 1
1 0
0 1
1 ے
ۑ ۑ ۑ
ې ܸൌ ۏ
ێ ێ ێ
ۍ^011001 11
(^0 1 0 1 ) 1 1
1 1
1 0
0 1
1 ے
ۑ ۑ ۑ
ې
Yes; with u 1 , u 2 , u 3 , u 4 , and u 5. v 1 , v 5 , v 2 , v 3 , and v 4.
Yes; with f(u 1 ) = v 1 , f(u 2 ) = v 4 , f(u 3 ) = v 3 , f(u 4 ) = v 2 , and f(u 5 ) = v 5.
- Determine whether the given graph is connected.
Ans: Connected Not connected (disjoint)
- Determine whether the given graph has an Euler circuit. Construct such a circuit when one exists. If no Euler circuit exists, determine whether the graph has an Euler path and construct such a path if one exists.
a. b. c.
Ans: a. The graph has exactly 2 vertices of odd degree c and f , then it has no Euler circuit, but has Euler path. One possible such path is f, a, e, f, b, a, d, b, c, d, e, c. b. Similarly as part a. c. Because all the vertices are of even degree, the graph has an Euler circuit. Try yourself to find one of these possible circuit.
- Can someone cross all the bridges shown in this map exactly once and return to the starting point?
Ans: Yes. For example, there exists an Euler circuit such as A, D, A, B, D, C, A.
- Determine whether the picture shown can be drawn with a pencil in a continuous motion without lifting the pencil or retracing part of the picture.
Necessary and sufficient conditions for Euler paths and circuits in directed graphs are
A directed multigraph having no isolated vertices has an Euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex are equal.
A directed multigraph having no isolated vertices has an Euler path but not an Euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex are equal for all but two vertices, one that has in-degree one larger than its out-degree and the other that has out-degree one larger than its in-degree.
Then, none of the above graphs has Euler circuit but still we can draw the graph in a continuous motion without lifting the pencil or retracing part of it if there is an Euler path.
a. No Euler path exists, since the graph has 3 vertices of different in-degree and out- degree; namely a , b , and d.
b. Yes there exists Euler path, since the graph has 2 vertices of different in-degree and out-degree but satisfy the conditions; namely a , and e. one of possible paths is { a, d, b, c, b, a, e, d, e, b, e, c, e }
