Eulerian Cycle (2A) Young Won Lim 5/25/18

Transcription

1 ulerian ycle (2)

2 opyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU ree ocumentation License, Version 1.2 or any later version published by the ree Software oundation; with no nvariant Sections, no ront-over Texts, and no ack-over Texts. copy of the license is included in the section entitled "GNU ree ocumentation License". Please send corrections (or suggestions) to This document was produced by using LibreOffice and Octave.

3 Path and Trail path is a trail in which all vertices are distinct. (except possibly the first and last) trail is a walk in which all edges are distinct. Vertices dges Walk may may (losed/open) repeat repeat Trail may cannot (Open) repeat repeat Path cannot cannot (Open) repeat repeat ircuit may cannot (losed) repeat repeat ycle cannot cannot (losed) repeat repeat ulerian ycles (2) 3

4 Simple Paths and ycles Most literatures require that all of the edges and vertices of a path be distinct from one another. ut, some do not require this and instead use the term simple path to refer to a path which contains no repeated vertices. simple cycle may be defined as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex There is considerable variation of terminology!!! Make sure which set of definitions are used... ulerian ycles (2) 4

5 Simple Paths and ycles Most literatures some other literatures trail circuit path cycle path cycle simple path simple cycle narrow sense path & cycle wide sense path & cycle ulerian ycles (2) 5

6 Paths and ycles e 1 e 2 e 3 e k v 0 v 1 v 2 v 3 v k One of a kind path v 0, e 1, v 1, e 2,, e k, v k path cycle cycle v 0, e 1, v 1, e 2,, e k, v k (v 0 = v k ) path v 0, e 1, v 1, e 2,, e k, v k (v 0 v k ) path cycle cycle v 0, e 1, v 1, e 2,, e k, v k (v 0 = v k ) Two different kinds ulerian ycles (2) 6

7 uler ycle Some people reserve the terms path and cycle to mean non-self-intersecting path and cycle. (potentially) self-intersecting path is known as a trail or an open walk; and a (potentially) self-intersecting cycle, a circuit or a closed walk. This ambiguity can be avoided by using the terms ulerian trail and ulerian circuit when self-intersection is allowed no repeating vertices repeating vertices repeating vertices repeating vertices ulerian ycles (2) 7

8 uler ycle visits every edge exactly once the existence of ulerian cycles all vertices in the graph have an even degree connected graphs with all vertices of even degree h ave an ulerian cycles non-repeating edges repeatable vertices circuit ulerian circuit : more suitable terminology ulerian ycles (2) 8

9 uler Path visits every edge exactly once the existence of ulerian paths all the vertices in the graph have an even degree except only two vertices with an odd degree n ulerian path starts and ends at different vertices n ulerian cycle starts and ends at the same vertex. non-repeating edges repeatable vertices trail ulerian trail : more suitable terminology ulerian ycles (2) 9

10 onditions for ulerian ycles and Paths n odd vertex = a vertex with an odd degree n even vertex = a vertex with an even degree # of odd vertices ulerian Path ulerian ycle 0 No Yes 2 Yes No 4,6,8, No No 1,3,5,7, No such graph No such graph f the graph is connected ulerian ycles (2) 10

11 The number of odd vertices # of odd vertices ulerian Path ulerian ycle 0 No Yes 2 Yes No # of odd vertices = 0 # of odd vertices = 2 ulerian ycle ulerian Path No ulerian Path No ulerian ycle ulerian ycles (2) 11

12 ulerian Graph ulerian graph : a graph with an ulerian cycle a graph with every vertex of even degree (the number of odd vertices is 0) These definitions coincide for connected graphs ulerian ycles (2) 12

13 Odd egree and ven egree ll odd degree vertices ll even degree vertices ulerian ycles (2) 13

14 uler ycle xample H H J G GHJK J G K a path denoted by the edge names K en.wikipedia.org ll even degree vertices ulerian ycles ulerian ycles (2) 14

15 ulerian ycles (2) 15 uler ycle xample en.wikipedia.org GHJK K J G H K J G H K J G H K J G H K J G H K J G H K J G H K J G H K J G H K J G H K J G H K J G H

16 uler Path and ycle xamples ulerian Path uerian ycle 1. uerian ycle 2. a path denoted by the vertex names ulerian ycles (2) 16

17 ulerian ycles of Undirected Graphs n undirected graph has an ulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component n undirected graph can be decomposed into edge-disjoint cycles if and only if all of its vertices have even degree. So, a graph has an ulerian cycle if and only if it can be decomposed into edge-disjoint cycles and its nonzero-degree vertices belong to a single connected component ulerian ycles (2) 17

18 dge isjoint ycle ecomposition H G J K ll even vertices uerian ycle dge isjoint ycles ulerian ycles (2) 18

19 ulerian Paths of Undirected Graphs n undirected graph has an ulerian trail if and only if exactly zero or two vertices have odd degree, and all of its vertices with nonzero degree belong to a single connected component. Here, the following definitions are used. Trail : walk without repeated edges. (closed or open) This definition includes trail (open walk) and circuit (closed walk) ll of which contain no repeating edges. ulerian ycles (2) 19

20 ulerian ycles of igraphs directed graph has an ulerian cycle if and only if every vertex has equal in degree and out degree, and all of its vertices with nonzero degree belong to a single strongly connected component. quivalently, a directed graph has an ulerian cycle if and only if it can be decomposed into edge-disjoint directed cycles and all of its vertices with nonzero degree belong to a single strongly connected component. ulerian ycles (2) 20

21 ulerian Paths of igraphs directed graph has an ulerian path if and only if at most one vertex has (out-degree) (in-degree) = 1, at most one vertex has (in-degree) (out-degree) = 1, every other vertex has equal in-degree and out-degree, and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph. ulerian ycles (2) 21

22 igraph ulerian ycle xamples 1:1 2:2 1:1 a b c 1:1 e 2:2 d a 2:2 e d c b 1:1 1:1 2:2 1:1 abcdbea edabcdcae ulerian ycles (2) 22

23 igraph ulerian Path xamples 2:2 2:1 a b e 1:1 d 1:2 dbadeab ulerian ycles (2) 23

24 Seven ridges of Königsberg The problem was to devise a walk through the city that would cross each of those bridges once and only once. ulerian ycles (2) 24

25 Seven and ight ridges Problems 7 bridges problem 8 bridges problem G H 4 ulerian Path HG ulerian ycles (2) 25

26 Nine and Ten ridges Problems 9 bridges problem 10 bridges problem G J G H H 5 ulerian Path 6 ulerian ycle HG HGJ ulerian ycles (2) 26

27 8 bridges ulerian Path G H G H G H G H G H G H G H G H G H ulerian Path HG ulerian ycles (2) 27

28 9 bridges ulerian Path G H G H G H G H G H G H G H G H G H G H ulerian Path HG ulerian ycles (2) 28

29 10 bridges ulerian ycle J G H J G H J G H J G H J G H J G H J G H J G H J G H J G H J G H ulerian ycle HGJ ulerian ycles (2) 29

30 leury s lgorithm To find an ulerian path or an ulerian cycle: 1. make sure the graph has either 0 or 2 odd vertices 2. if there are 0 odd vertex, start anywhere. f there are 2 odd vertices, start at one of the two vertices 3. follow edges one at a time. f you have a choice between a bridge and a non-bridge, lways choose the non-bridge 4. stop when you run out of edge ulerian ycles (2) 30

31 ridges bridge edge Removing a single edge from a connected graph can make it disconnected Non-bridge edges Loops cannot be bridges Multiple edges cannot be bridges ulerian ycles (2) 31

32 ridge examples in a graph ulerian ycles (2) 32

33 ridges must be avoided, if possible bridge f there exists other choice other than a bridge The bridge must not be chosen. bridge ulerian ycles (2) 33

34 leury s lgorithm (1) ulerian ycles (2) 34

35 leury s lgorithm (2) : bridge : bridge :bridge : bridge : chosen : chosen :chosen no other choice : chosen no other choice ulerian ycles (2) 35

36 leury s lgorithm (3) : bridge : chosen no other choice : bridge : chosen no other choice ulerian ycles (2) 36

37 egree of a vertex the degree (or valency) of a vertex is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex v is denoted deg(v) the maximum degree of a graph G, denoted by Δ(G) the minimum degree of a graph, denoted by δ(g) Δ(G) = 5 δ(g) = 0 n a regular graph, all degrees are the same 1 a g 0 3 b f 2 3 c e 5 2 d ulerian ycles (2) 37

38 Regular Graphs a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. ulerian ycles (2) 38

39 Handshake Lemma The degree sum formula states that, given a graph G = ( V, ) 1 a 3 b 3 c 2 d This statement (as well as the degree sum formula) is known as the handshaking lemma. g 0 f 2 e 5 deg(a) = 1 deg(b) = 3 deg(c) = 3 deg(d ) = 2 deg(e) = 5 deg(f ) = 2 deg(g) = 0 16 = 8 2 = 16 ulerian ycles (2) 39

40 dding odd vertex ulerian ycles (2) 40

41 The number of odd vertices ven vertices : {x 1, x 2,,x m } Odd vertices : {y 1, y 2,, y n } S = deg(x 1 ) + deg(x 2 ) + + deg(x m ) T = deg( y 1 ) + deg( y 2 ) + + deg( y n ) deg(x i ) : even deg( y i ) : odd S = even + even + + even T = odd + odd + + odd S : even S+T : even T : even = n odd numbers n : even in any graph, the number of vertices with odd degree is even. ulerian ycles (2) 41

42 uler ycle ny connected graph with even degree vertices n uler cycle proof by induction on the number of edges in G connected graph G with even degree vertices only and k edges (k < n) n uler cycle ssume this is true connected graph G with even degree vertices only and n edges n uler cycle Then this holds true Johnsonbough, iscrete Mathematics ulerian ycles (2) 42

43 uler ycle ny connected graph with even degree vertices which has n edge n uler cycle ny connected graph with even degree vertices which has n-1 edge ny connected graph with even degree vertices which has n-2 edge n uler cycle n uler cycle ny connected graph with even degree vertices which has 2 edge ny connected graph with even degree vertices which has 1 edge n uler cycle n uler cycle Johnsonbough, iscrete Mathematics ulerian ycles (2) 43

44 uler ycle ase ases n = 0 edge n = 1 edge all even degree vertices n = 2 edge an uler cycle Johnsonbough, iscrete Mathematics ulerian ycles (2) 44

45 uler ycle decrease the number of edges by one connected graph G with even degree vertices only and n edges (k < n) e 1 e 2 v 1 v 2 v 3 all even degree vertices P: a path from v to v 1 connected graph G' with even degree vertices only and n-1 edges (k < n) P': a portion of the path P that are in G' e v 1 v 2 v 3 all even degree vertices Johnsonbough, iscrete Mathematics ulerian ycles (2) 45

46 uler ycle a path from v to v 1 ase 1: P ends at v 1 ase 2: P ends at v 2 ase 3: P ends at v 3 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P P P v v v v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P ' P ' P ' v v v Johnsonbough, iscrete Mathematics ulerian ycles (2) 46

47 uler ycle H J G GHJK K en.wikipedia.org ulerian ycles (2) 47

48 uler ycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components ulerian ycles (2) 48

49 uler ycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component ulerian ycles (2) 49

50 uler ycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components ulerian ycles (2) 50

51 uler ycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component ulerian ycles (2) 51

52 References [1] [2]