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To determine If a graph contains a circuit that starts and ends at a vertex v, does the graph contain a simple circuit that starts and ends at v? Why?

To determine If a graph contains a circuit that starts and ends at a vertex v, does the graph contain a simple circuit that starts and ends at v? Why?

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Expert Solution To determine If a graph contains a circuit that starts and ends at a vertex v, does the graph contain a simple circuit that starts and ends at v? Why? Answer to Problem 48ES Yes Explanation of Solution Given information: A graph contains a circuit that starts and ends at a vertex v. Calculation: Let vejvjezv.v-Je,v be a circuit that starts and ends at vertex v in some graph G (for some positive integer n ). If the circuit vejvje2v2..Vn–1enV contains no repeated vertices, then the circuit vej vie2v2...Vn–1Env is a simple circuit as well. If vejvjezv2..Vn-1enV contains some repeated vertex Vị = vị (1 < ij <n- 1 with i + j), then we remove the part of the circuit between these two vertices. the resulting path will still be a circuit, because the circuit will still contain at least one edge and will still contain no repeated edges. repeat this procedure until the resulting circuit contains no more repeated vertex and the resulting circuit then is a simple circuit. We then note that the graph needs to contain a simple circuit that starts and ends at v. Vi+1 ..... Vn-1 Vi+1
Transcribed Image Text:Expert Solution To determine If a graph contains a circuit that starts and ends at a vertex v, does the graph contain a simple circuit that starts and ends at v? Why? Answer to Problem 48ES Yes Explanation of Solution Given information: A graph contains a circuit that starts and ends at a vertex v. Calculation: Let vejvjezv.v-Je,v be a circuit that starts and ends at vertex v in some graph G (for some positive integer n ). If the circuit vejvje2v2..Vn–1enV contains no repeated vertices, then the circuit vej vie2v2...Vn–1Env is a simple circuit as well. If vejvjezv2..Vn-1enV contains some repeated vertex Vị = vị (1 < ij <n- 1 with i + j), then we remove the part of the circuit between these two vertices. the resulting path will still be a circuit, because the circuit will still contain at least one edge and will still contain no repeated edges. repeat this procedure until the resulting circuit contains no more repeated vertex and the resulting circuit then is a simple circuit. We then note that the graph needs to contain a simple circuit that starts and ends at v. Vi+1 ..... Vn-1 Vi+1
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