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By Rainer E. Burkard, Ulrich Derigs (auth.)

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EXTERNAL SUBROUTINES : SUBROUTINE SSORT 6. DERIGS G. O) GOTO 120 SM(I ) -0 STACK(IT) -I IT =IT+l CONTINUE IS -I 00 50 53 55 110 120 31 G C G *** GROWING ALTERNATING TREES FROM EACH UNSATURATEO VERTEX 200 GONTINUE =STACK(IS) NS -BASIS (NS) NBS TMA(NBS) =0 =INOEX(NS) I -NS+I II -INOEX(II )-1 II NI .. NK2) IZ =IZ+! , 10X, 15HNUMBER OF EDGES, 14) FORMAT(/! ,46RERROR: TRE VERTEX DEGREES DO NOT SUM UP TO 2*M) STOP END 35 CARDINALITY MATCHING PROBLEM NUMBER NUMB ER OF NODES NUMBER OF EDGES LI ST OF EDGES LIST 2 12 l2 3 15 l5 17 l7 19 19 24 27 4 6 8 5 4 5 5 6 6 6 6 7 8 10 lD 12 l2 13 l3 13 l3 14 l4 14 l4 14 l4 14 l4 1l 5 15 l5 l6 16 l8 18 18 l8 20 21 2l 21 2I 21 22 22 23 23 24 25 26 lO 10 6 9 7 8 9 lO 10 11 II 9 11 II 13 l3 l4 14 15 l5 16 l6 18 l8 20 21 2l l7 17 20 17 l7 2l 21 25 21 2l 22 23 25 23 25 24 26 27 26 27 27 46 36 OPTIMAL lIATCHING 1 12 2 3 4 S 6 19 8 10 9 0 8 9 10 3 S 4 12 13 1 15 7 11 11 7 14 20 15 16 17 18 13 17 16 21 19 20 2 14 21 18 22 23 24 2S 25 24 23 22 26 27 27 26 CARDINALITY OF TRE OPTIMAL MATCRING 13 10 11 26 Fig.

This circle induces a subgraph which is determined by backtracing from both outer vertices v and w to the root s. Then this subgraph is shrunken to a so called pseudonode. This pseudonode is made an outer vertex of the tree. With every augmenting path in the shrunken graph we can associate an augmenting path in the original graph and vice versa. Case 4) v is only connected with inner vertices. During the tree construction procedure the neighbourhood of outer vertices is examined successively following the above mentioned scheme.

During the tree construction procedure the neighbourhood of outer vertices is examined successively following the above mentioned scheme. This results either in the detection of an augmenting path in case 1) ar in the situation that case 4) holds for each outer vertex. In that case it can be shown that na augmenting path starting from s does exist. The subgraph induced by the labeled vertices can now be excluded from further inspection. If there is more than one unsaturated vertex left another alternating tree is constructed.

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