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Program EulerPath_Algorithm;
(**************************************************************************}
{ An Euler path (pronounced 'oiler') in a graph G is a path that uses each }
{ arc of G exactly once and can exist in a connected graph iff there are }
{ either no nodes whose degree is odd or exactly two nodes whose degree is }
{ odd. For the case of no nodes posessing an odd degree, the path can }
{ begin at any node and will end there; for the case of two nodes posessing}
{ an odd degree, the path must begin at one odd node and end at the other. }
{--------------------------------------------------------------------------}
{ Released to the public domain by Natural Systems. This routine may be }
{ freely distributed. }
{--------------------------------------------------------------------------}
{ Natural Systems is the producer of DATA STRUCTURES (DS), a highly }
{ versital library of the most critical structures--such as graphs--for }
{ all Borland compilers supporting OOP. }
{ }
{ Euler's, as well as the Dijkstra, Warshall, Bellman-Ford, Floyd's, }
{ Kruskal's (and many, many more) methods, are available in DS. And }
{ that's just one class--there's too many to list... (Of course, because }
{ of its' object oriented nature, there are no restrictions or side- }
{ effects. Just good solid code!) }
{ }
{ Contact Natural Systems on CompuServe @ [71301,1221], by phone/FAX @ }
{ (617) 232-6951, or by post @ PO Box 968 / Brookline, MA / 02146 to get }
{ more information about or, better yet, purchase DS. MasterCard & Visa }
{ Accepted. }
{**************************************************************************)

{$IfDef Windows}Uses WinCRT;{$Else}Uses CRT;{$EndIf}

Const
n = 5;

Var
M : Array [1..n, 1..n] of Word;

Function IsOdd(x:LongInt) : Boolean;
{**************************************************************************}
Begin
If ( (x MOD 2) = 0 ) Then
IsOdd := False
Else
IsOdd := True;
{EndIf}
End;
{EndFunction}

Procedure EulerPath;
{**************************************************************************}
{ Determins whether an Euler path exists in a connected graph with }
{ adjacency matrix M. }
{**************************************************************************}
Var
Total : Integer; { Number of odd nodes found so far }
Degree : Integer; { Degree of a node }
i, j: Word; { indices }

Begin
Total := 0;
i := 1;

While (Total <= 2) AND (i <= n) Do
Begin
Degree := 0;

For j := 1 to n Do
Degree := Degree + M[i, j];
{EndFor}

If IsOdd(Degree) Then
Inc(Total);
{EndIf}

Inc(i);
End;
{EndWhile}

If Total > 2 Then
Write('No Euler path exists')
Else
WriteLn('Euler path exists');
{EndIf}
End;
{End EulerPath}

Var
i, j : Word;

Begin
M[1,1] := 0; M[1,2] := 1; M[1,3] := 0; M[1,4] := 0; M[1,5] := 0;
M[2,1] := 0; M[2,2] := 0; M[2,3] := 1; M[2,4] := 0; M[2,5] := 0;
M[3,1] := 1; M[3,2] := 0; M[3,3] := 0; M[3,4] := 1; M[3,5] := 0;
M[4,1] := 0; M[4,2] := 0; M[4,3] := 0; M[4,4] := 0; M[4,5] := 0;
M[5,1] := 1; M[5,2] := 0; M[5,3] := 1; M[5,4] := 0; M[5,5] := 0;

For i := 1 To n Do
Begin
For j := 1 To n Do
Write(M[i, j]:3);
{End For}

WriteLn;
End;
{End For}

EulerPath;

WriteLn;
M[4,5] := 1;

For i := 1 To n Do
Begin
For j := 1 To n Do
Write(M[i, j]:3);
{End For}

WriteLn;
End;
{End For}

EulerPath;

ReadLn;
End.