Vdumc: 2, number 6 lNFORMATION PROCESSING LEITERS December 1976

A NOTE QN DEGREECONSTRAINED STAR SIJBGRAPHS OF BIPARTITE GRAPHS *

Harold N. GABOW Lbgwtmertt o/Cumptm science, University of Colorado, Boulder, Coioradc. 88309, (;&I

Received 28 October 1975, revised version re4ved 16 September 19’16

AM&Y& of &orithms, formal languages, space complexity, bracket languages, bipartite graph, star. degree-constrained sub$rq?k, matding.

Much work has been done on finding degree-con- e& tig~@~ of b@rtM gragks f5,6), and tie sP%al CIIse bf matchings 141. In this note we inves-

-1 s&ion X?f%Mn%diate generality , “j%~ sub- grapbr”. We indicate ho* these subgraphs relate to matchings, and how, in some case~s. they can be con- structed tff&38ntly.

We fit @%&e some terms-and notation. Standard teims not d&&neB h4+i+e are in [3]. Thrwghout this note, w$dt oursldvibs to a bipartite graph G = (S, T); here S’ a& Tare the two mta that.p;rtitIon the ver- ticm TM degree of a’vertblt-U is dertoted a(v), or c3(U,O)ifti&3@+iC~ot’apparePt.

LetF be anfisubgraph. A path P is dternuting (wiur rtspect to F) if it is simple (i.e., no vertex is re- peated), and its edges rue altexnatively in F and in G-E P is nctgmentiqg if it fs atternating, the fiit and

* This work was partially supported by the National Science Foundation under Grant GJ-099’72.

last edges are in G-F, and the fit and last vez&es are unsriturated. If P is augmenting, thz edges in E 8 P = F U P - F n P form an fisubgraph containing onr3 more edge than F. if F is not a maximum fisubgaph,

an aUgme!NizIg path exists. ‘I%s follows from network fldw -&eory [2f.

A S&U is a tree consisting of a root and a rwnbex of SW. We defme an f-stcn S&_-/Z as ar! P_subgraph F that consists of vertexdisjoint starts. 4 m&mum f- sttw subgraph contains the greatest number ot &&es possible. If G = (S, T), then F is roared in T if tke

root of each star is in T. An fetar subgraph rooted in 2% equivalent to a g-subgraph, wheTe Ir(u) = 1 for u E S and g(u) = flu) for u E 7’. So an f-star subgraph rooted 31 T is a speciaI ?ztie of a degree-constrained subgraph. In particular, the results on augmenting

Paw VP&* We begin by showing a relationship between f-star

subgraphs and match@s.

Theorem 1: Let F be a maxinnum f-star subgrapfi root- ed in T, covering the greatest number of vertices pas- sible. Then a maximum matdhing M on F in a maxi- mum matching on G. Proof: First observe M is constructed by choosing one edge from each star of F. NOW suppose M is not maxi- mum, so it has an augmenttig path P. We derive a con- trzdiction below.

Let the first and last vertices off be of S and w E T. Vertex w is not covered by M, SO it is not CW- ered by F. ‘l’h& implies v meets some edge w E PX %or otherwise, v is cot saturated by F (WC assume

f&me 5, number 6 P~FORhkATKiN PROCESSING LETTERS Decembar 1976

Volume 5, number 6 INFORMATION PRWESSING LETTERS Wmtwr 1976

Theorem, then all graphs constructed do too. So the f-star subgraph is found in the time bound of Corol-

W2 No\ r we consider general f-star subgraphs of a bi-

partite with a perfect matchin& 4 maximum ’ matching k an f%tar sulqraph that covers all vertices;

it can be found tn OQ!P Y + V) time [4]. The follow ing &o&m improves this timer bound whenf, 2.

plvmdurrc;

8 camneat This a@Mm finds an f-star subgraph 8 cover%g ail vertices of a bipartite graph

C = (S, T) with a perfect matching;

beeEn A 1. construct H, a maximum fistar subgraph rooted in

T; comment use Dinic’s algorithm; 2. construct M, a maximum matching on AT, comm4M

choose an edge from each star of&, 3. construct F, a maximum$star subgraph rooted in

& covering alI vetices covered by M, comment use JXnic’s algorithm, starting with N as the &tial sub graph;

4. f~ each vertex uE Tdo aesin

5. ktA be~~s~ofelledgesuwEH,wherewis not covered by F,

6. ifR has at least one edge then

7. let B be the star in F con&i&g o; 8. ifS has.at least two edges then

delete the edge of B containing v from F, 9. add A to FV

=k @M

@@d;

VIWMIHQ 3: Let G be a bipartite graph with a perfect rna&Mng~W flu) 9 2 for a@ vertices 0, Algorithm C constructs anf-star subgraph F covering all vertices, inqEla$v)ths.

Roof, First we show F is constructed correctly. Line 1 constructs a maximum @.ar subgraph H rooted in T. H covers all vertices in S, Bince C has a perfect matching. Similarly, line 3 constructs an f’tar sub- graph F covering all vertices in T. F also covers ah ver- tices coveled by M, assuming M is the initial sub, yapu in Dinic’s &orithm. For IXnic”s algorithm work by repeatedly augmenting M, and in an aument, nti cov-

ered vertex gets uncovered. So after line 3, F covers

& vertices evaept possibiy vertices w E S not CC rered by M. TcJc= w 410~ the loop in lines 4-9 covers thexr

vertii 2s. Vertex w is in a star 0fH. So at some point, lines A

4-5 choose u and A 30 edge VW is in star A. Vertex tt is covered by exactly one ed ,ofastarBinF.If R contains no edge besides W, then line 9 adds .A to F, this covers w, without violating any contraints. [In particular, u meets at most f(u) edges, since u meets fewer than fl’u) edges of A] . If B has at least two ed- ges, then line& deletes W, and line 9 adds A; this COW ers w, without uncovering u or violating any constraints. So in both cases, F is modified correctly to cover W. After the loop, F covers ah vertices. So algorithm C is correct.

Now consider the run time. Lines 1 and 3 use O@ lIq v) time, by Corollary 2. Lines 2,4--9 use O(E + v) = O@) time, with the appropriate choice of data structues. So the total time is O(E log y) . QED

In the special case where ,f(u) = 2 for dl vertices u, algorithm C covers all vertices by a collection of vertex- disjoint paths of Iength 1 or 2.

The author thanks the referees for their susestions, and in particular, for observing that Theorem I was originally stated incorrectiy .

Ref-s

[ 1 ] E.A. Dinic, Algorithm for solution of a problem of maxi- _ - ~UITI flow in a network with power estimation, Sov. path, Dokl. 11, S (1970) 1277-1280.

(2) LX Ford and D.R. Fulkerson, Flows in Networks (Prin- ceton Wniversity Press, Princeton, N.J., 11962).

[3] F. Harary, Graph Theory (Addison-Wesley, Reading, Mass., 1969).

14) J.E. Mopcroft and R.M. Karp, An nsf2 algorithm rbr maximum matchings i;l bipartite graphs, SIAM J. Comput. 1(1973) 225-231.

[S ] R.E. Tarjm, Testing graph connectivity, Proc. 6th Annual ACM Symp. on Theory on Computing, Se*Lttle, Washing ton, 1974,185-193.

(6) RJ. Urquhart, Degree-constrained subgraphs of linear graphs, Ph.D. Diss., University of

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