ALGORITHMS FOR FINDING AN EULERIAN TRAIL

Transcription

1 THE 1 ST INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS AND INFORMATICS AT UNIVERSITIES 2001 Section of didactic of mathematics and informatics, pp ALGORITHMS FOR FINDING AN EULERIAN TRAIL MILAN POKORNÝ Trnava, Trnava University, Faculty of Education Abstract: The trail T of the graph G is called Eulerian one, if it is closed and contains all the vertices and edges of the given graph G. In the present paper we describe two algorithms for finding the Eulerian trail of the graph. We also describe our software which can be used in teaching methods. This part of the theory of graphs is suitable for different types of schools. In mathematics it is very often possible to run into a problem to draw the given picture in one move and go through out each line only once (through the intercepts lines it is possible to go arbitrarily many times). It is known that there are quite difficult pictures on one side that can be drawn in one move and on the other side there are very simple pictures that cannot be drawn in one move. The problem has been evolved to find necessary and sufficient condition for drawing the picture in one move. This condition has been known in mathematics since the 18 th century. In 1736 an important mathematician L. Euler solved the following problem. The river Pregel, that flowed through the town Königsberg, formed two islands and so the town was divided into four parts joined together with seven bridges (see picture No.1). The problem was to design a circular walk around the town in the way to cross over the brigde only once. 364

2 L. Euler not only proved that such walk didn t exist but also he avolved the theory of solving similar problems. [3], [5], [1] The above mentioned problem of drawing the picture in one move that belongs into the field of funny mathematics has also got a practical utilization for optimalization of some Picture No. 1 situations related to distribution of goods etc. The problem of drawing picture in one move only can be transformed into the theory of graphs. Every intercept of two or more lines (or initial and end point of a line) is considered as a vertex of the graph and every line connecting two of these points is considered as an edge of the graph determined by these points. All basic definitions and terms such as graph, directed graph, connected graph, trail, etc. from the theory of graphs can be found in [3], [1], and [5]. In this article we deal with only simply undirected graphs without loops and multiple edges. All of mentioned terms and results can be extended into directed graphs, mixed graphs or multigraphs. The trail T of the graph G is called Eulerian trail, if it is closed and contains all the vertices and edges of the given graph G. Assuming the preceeding definition without the closing condition the trail is called open Eulerian trail. Graph G is called Eulerian graph if it contains the Eulerian trail. Eulerian graphs are characterized by the following theorem: Theorem: (see [3, Theorem 7.1]) For the arbitrary connected graph (multigraph) G, the following statements are equivalent: 1. the graph G is the Eulerian one, 2. a degree of every vertex of the graph G is even, 3. the set of edges of the graph G can be decomposed into simple cycles. The following simple algorithm is based on the idea of previous theorem to find Eulerian trail (see [4]). If we suppose the graph G is Eulerian one we can start form the arbitrary vertex v 0 of the graph G and ride over the ungone edges of the graph G. This riding will stop in the vertex v 0 because the degree of every vertex of the graph G is even. In this way we have formed the v 0 -v 0 trail T. If E(T)=E(G) then T is Eulerian trail. Otherwise we select a nontrivial component C G-E(T). Because the graph G is connected, T and C have a common vertex w. We will form the nontrivial w-w trail Q of the graph C by the same way as we have formed the v 0 -v 0 trail T. If we insert the trail Q into the trail T, we will obtain the 365

3 v 0 -v 0 trail T which is longer than the trail T. Because the graph G is the finite one the extention of the trails like that leads to Eulerian trail of the graph G. Edmonds and Johnson (1973) (see [2]) designed more effective algorithm on the base of Tarry s algorithm for the research of labyrinth. In the graph G we can choose the arbitrary vertex v 0 and then form a walk according to the following rules: - we are allowed to go over the edge in the same direction only once - at the point of leaving the arbitrary vertex we prefer the edges in this order: 1. non-gone over edges yet 2. gone over edges only in the direction to this vertex besides the first coming edge 3. the first coming edge Alongside it we remember the direction of going over the edges and for every vertex v v 0 of the graph G also the first coming edge into v. We try to remember the edges in that order how they are gone over the second time. This sequence of the edges is called feedback one and it is the searched Eulerian trail. (see [4, page 262]) Introduced method is a special case of Tarry s algorithm and if it is impossible to go on every edge of the component containing the vertex v 0 is gone over in both directions only once. This algorithm can be realised by the following programme (written in Turbo Pascal): uses crt; var f:text; {we suppose that information about a graph are in the text file with structure: the 1 st line number of vertices, the 2 nd line number of edges, the 3 rd and the 4 th line the vertices of the first edge, the 5 th and the 6 th line the vertices of the second edge, etc.} first_coming:array[1..100] of integer; {we remember the edge of the first coming for every vertex} edges:array[1..100,1..100] of byte; {this array contains information about the edges: 0-edge doesn t exist, 1- non-gone over edge, 2- gone over edge, 3- gone over edge, but in the reverse direction, 4- gone over edge in both directions} connected:array[1..100] of boolean; {if the vertex is connected with the vertex number one} i,j,ii,jj,vertex_number,edges_number,ph:integer; active_vertex,goal,gone_edges:integer; connect,change:boolean; c:char; function next_not_gone(i:integer):integer; {it finds the edge which started in the vertex i and we haven t gone over it yet and returns the number of the second vertex of this edge} var pom,pi:integer; pom:=0; for pi:=1 to vertex_number do if edges[i,pi]=1 then pom:=pi; next_not_gone:=pom; 366

4 function next_gone (i:integer):integer; {it finds the edge which started in the vertex i and we have already gone over it and returns the number of the second vertex of this edge } var pom,pi:integer; pom:=0; for pi:=1 to vertex_number do if (edges[i,pi]=3)and(pi<>first_coming[i]) then pom:=pi; next_gone:=pom; gone_edges:=0; {number of edges which are gone in both directions} for i:=1 to 100 do for j:=1 to 100 do edges[i,j]:=0; {reading information about the graph from the text file graph.pas} assign(f,'graph.pas');reset(f); {we suppose that text file is correct} readln(f,vertex_number); {number of vertices} for i:=1 to vertex_number do first_coming[i]:=0; readln(f,edges_number); {number of edges} for i:=1 to edges_number do {reading information about edges} readln(f,ii); readln(f,jj); edges[ii,jj]:=1; edges[jj,ii]:=1; {we check if the graph is connected} {connected[i] is true if there exist a path from vertex 1 to the vertex i} for i:=2 to vertex_number do connected[i]:=false; connected[1]:=true; repeat change:=false; for i:=1 to vertex_number do if connected[i] then for j:=1 to vertex_number do if (edges[i,j]>0) and (not connected[j]) then connected[j]:=true; change:=true; until (not change); {if connected[i] is false for some vertex i 2..vertex_number then the graph is not connected and programme will be finished} connect:=true; for i:=2 to vertex_number do if (not connected[i]) then connect:=false; if (not connect) then writeln('your graph isn t connected.'); readln; halt; 367

5 {we check if the degree of each vertex is even} for i:=1 to vertex_number do ph:=0; {number of edges incoming to the vertex i} for j:=1 to vertex_number do if edges[i,j]>0 then inc(ph); {if ph is odd for some vertex, programme will be finished} if (ph mod 2 = 1) then writeln('the degree of the vertex ',i,' is odd!'); readln; halt; {now we can start to find an Eulerian trail, because graph is connected and the degree of each vertex is even} active_vertex:=1; {we started in the vertex 1} repeat {first we are trying to go over the edges we haven t gone yet} if next_not_gone (active_vertex)>0 then goal:=next_not_gone (active_vertex); edges[active_vertex,goal]:=2; edges[goal,active_vertex]:=3; if first_coming[goal]=0 then first_coming[goal]:=active_vertex; active_vertex:=goal; end {then we chose the edges we have gone over only in the direction to the vertex active_vertex, but not the edge of the first coming} else if next_gone (active_vertex)>0 then goal:=next_gone(active_vertex); edges[active_vertex,goal]:=4; edges[goal,active_vertex]:=4; if first_coming[goal]=0 then first_coming[goal]:=active_vertex; writeln(active_vertex,'-',goal); active_vertex:=goal; inc(gone_edges); end {finally we go over the edge of the first coming} else goal:=first_coming[active_vertex]; edges[active_vertex,goal]:=4; edges[goal,active_vertex]:=4; if first_coming[goal]=0 then first_coming[goal]:=active_vertex; writeln(active_vertex,'-',goal); active_vertex:=goal; inc(gone_edges); end until (edges_number=gone_edges); Readln; end. If we want to use this algorithm in teaching at the different types of schools, we should solve some problems. We should start teaching with simple graphs and then use obtained 368

6 knowledge for more difficult graphs. However if the graph we want to draw in one move is too simple, students mostly draw it experimentally, without using this algorithm. That s why we decided to produce software that will serve to understand and practice this algorithm. All this software you can find on the web pages of the Department of mathematics and informatics of Pedagogical Faculty of Trnava University. The programme ukaztah shows the graph in the graphic mode. It simulates the algorithm of Edmonds and Johnson using colours. It stops after each step. Execution of the next step can be activated by pressing a key by user. Explanation in more details for all programs can be found in the file citajma.doc. The programme najditah can serve for teaching of finding Eulerian trail. The user can see the whole situation in the way as if he just walked along the graph. So he cannot see all graph. He can see only his local part, that is the vertex, in which he stands and the edges incoming to this vertex. Different groups of the edges (non-gone over, gone in both directions, gone in the direction to the vertex, gone in the direction from the vertex, the first coming edge) are displayed using different colours. If the user s decision in the vertex is wrong, the programme warns him about his mistake and makes him correct a mistake himself. At the end the programme displays all graph and shows user s going over the graph step by step. The difference between the programmes skusanie and najditah can be seen if user s decision in a vertex is wrong. Programme skusanie ignores user s mistakes and at the end it displays all graph and shows user s going over the graph step by step. Now user can see if he really found an Eulerian trail or not and if he did it according to Edmonds and Johnson s algorithm. If he made a mistake in some vertex, this part of programme warns him about it. Programme urobgraf enables to user to create a graph in graphic mode (he can create a graph as a text file following instructions in the file citajma.doc). This is more comfortable because he can see all graph. He can add and delete vertices and edges. He can also take some vertex and move it. Edges incoming to this vertex are moved automatically, of course. That s why this programme can be used in teaching isomorphic graphs, but it isn t our purpose. In this article we deal with only simple undirected graphs, but similar problems we can solve in directed graphs and mixed graphs (see [4]). If the Eulerian trail of the graph doesn t exist, we often find Eulerian walk (some edges can be occured more times). In practical life we often find the shortest walk, especially for evaluated graphs. References [1] BOSÁK, J.: Grafy a ich aplikácie. Alfa, Bratislava [2] ČUPR, K.: Geometrické hry a zábavy. Cesta k vědení, Praha [3] HARARY, F.: Graph theory. Reading, Addison Wesley, Moskva 1973 (Russian translation). [4] PLESNÍK, J.: Grafové algoritmy, Veda, Bratislava [5] SEDLÁČEK, J.: Úvod do teorie grafů. Academia, Praha