An Algorithm Computing the Core of a Konig-Egervary Graph

An Algorithm Computing the Core of a K¨ onig-Egerv´ ary Graph V adim E. Levit 1 and Eugen Mandrescu 2 1 Ariel Universit y Cen ter of Samaria, ISR AEL levitv@ari el.ac.il 2 Holon Institute of T echnology , ISR AEL eugen m@hit.ac.i l Abstract. A set S of vertices is indep endent (or stable ) in a graph G if no tw o vertices from S are adjacent, and α ( G ) is the cardinalit y of a largest (i.e., maximum ) indep end ent set of G . G is called a K¨ onig-Egerv´ ary gr aph if its order equals α ( G ) + µ ( G ), where µ ( G ) den otes the size of a maximum matching. By core( G ) we mean the inters ection of all maxim um indep endent sets of G . T o decide whether core( G ) = ∅ is know n to b e NP -hard [1]. In th is pap er, we present some p olyn omial time algorithms find ing core( G ) of a K¨ onig-Egerv´ ary graph G . Key words: maximum independ ent set, maxim um matching, core 1 In tro duction Throughout this pap er G = ( V , E ) is a finite, undirected, lo opless and without m ultiple edg es graph with vertex set V = V ( G ) of cardina lit y | V ( G ) | = n , and edge s et E = E ( G ) of cardina lit y | E ( G ) | = m . If X ⊂ V , then G [ X ] is the subgraph of G spanned by X . By G − W we mea n the subg raph G [ V − W ], if W ⊂ V ( G ). The neighborho o d of a vertex v ∈ V is the set N ( v ) = { w : w ∈ V and v w ∈ E } , while N ( A ) = ∪{ N ( v ) : v ∈ A } and N [ A ] = A ∪ N ( A ) fo r A ⊂ V . A se t S ⊆ V ( G ) is indep endent if no t wo vertices from S are adjacent; b y Ind( G ) we mean the set o f all the indep endent sets of G . An indep endent set o f maximum size will b e refer red to as a maximum indep endent set of G , and the indep ende nc e numb er of G is α ( G ) = ma x {| S | : S ∈ Ind( G ) } . In the se q uel, the family { S : S is a maximum indep endent set of G } is deno ted b y Ω ( G ). A matching in a g raph G = ( V , E ) is a set M ⊆ E such that no t wo edge s of M s hare a common vertex. A matc hing of maximum cardinality µ ( G ) is a maximum matching , and a p erfe ct matching is one cov ering a ll vertices o f G . It is known that α ( G ) + µ ( G ) ≤ | V ( G ) | . If α ( G ) + µ ( G ) = | V ( G ) | , then G is called a K¨ onig-Egerv´ ary gr ap h (Deming [4], and Sterb oul [24]). It is easy to see that if G is a K¨ onig-E gerv´ ary graph, then α ( G ) ≥ µ ( G ), a nd that a gra ph G having a per fect matc hing is a K¨ onig-Eger v´ ary gr aph if and only if α ( G ) = µ ( G ). 2 Levit and Mandrescu K¨ onig-Eger v´ ary graphs were in vestigated in several papers , among w e quo te [3,10,13,15,16,19,20,23], and g eneralized in [2,2 1]. According to a ce lebrated result o f K¨ onig [9], and Eg erv´ ary [6], every bipartite graph is a K¨ onig-E gerv´ ary gr aph. This class includes non- bipartite graphs as well (see, for instance, the gr a phs H 1 and H 2 in Figure 1). ✇ ✇ ✇ ✇ ❅ ❅ ❅ H 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇       H 2 ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ H 3 Fig. 1 . Only H 3 is no t a K¨ onig– E gerv´ ary graph, a s α ( H 3 ) + µ ( H 3 ) = 4 < 5 = | V ( H 3 ) | . A characterization o f K¨ onig-Eg erv´ ary graphs has b een fo und independently by Deming [4] and Sterb oul [24]. Recently , it has been presented a forbidden sub- graph characterization of K¨ onig- E gerv´ ary graphs [11]. Other characterizations of K¨ onig-Eger v´ ary gr aphs can b e found in [1 2,17,18]. Theorem 1. [7], [4] Given a gr aph G and a maximum matching of G , one c an test whether G is a K¨ onig-Egerv´ ary gr aph in time O ( m + n ) . Theorem 2. [25] Given a gr aph G , one c an find a maximum matching in time O ( m • √ n ) . As a consequence of Theorems 1, 2 o ne can deduce the following. Corollary 1. Given a gr aph G , one c an che ck in t ime O ( m • √ n ) whether G is a K¨ onig-Egerv´ ary gr aph. Let us rec a ll that core( G ) = ∩{ S : S ∈ Ω ( G ) } , [14]. Prop ositio n 1. [14] F or a c onne cte d bip artite gr ap h G = ( A, B , E ) of or der at le ast two, the fol lowing assertions ar e true: (i) α ( G ) > | V ( G ) | / 2 if and only if | c o re( G ) | ≥ 2 ; (ii) α ( G ) = | V ( G ) | / 2 if and only if | c o re( G ) | = 0 and A, B ∈ Ω ( G ) . Notice that Prop ositio n 1 ( i) is not true for non-bipartite K¨ onig-E gerv´ ary graphs; e .g ., the g raph G 2 from Fig ure 2 . Theorem 3. [14] F or a c onne cte d K¨ onig-Egerv´ ary gr aph G = ( V , E ) of or der at le ast two, the fol lowing assertions ar e true: (i) α ( G ) > | V ( G ) | / 2 if and only if | c o re( G ) | > | N ( core ( G )) | ≥ 1 ; (ii) α ( G ) = | V ( G ) | / 2 if and only if G has a p erfe ct matching. It is known, [1], that if G has no isola ted vertices, then α ( G ) > µ ( G ) ⇒ | core( G ) | > α ( G ) − µ ( G ) . Moreov er, if G is a connected g raph satisfying 3 µ ( G ) < | V ( G ) | , then G has | core( G ) | ≥ 2, [1]. Core of a K¨ onig-Egerv´ ary Graph 3 ✇ ✇ ✇ ✇ ✇ ✇ ✇ u v G 1 ✇ ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅   ❅ ❅ ❅ a b c G 2 Fig. 2 . G 1 has α ( G 1 ) = 4 > | V ( G 1 ) | / 2 and cor e( G 1 ) = { u , v } , while the gra ph G 2 has α ( G 2 ) = 3 = | V ( G 2 ) | / 2 a nd | core( G 1 ) | = | { a, b, c }| ≥ 2. Theorem 4. [1] The pr obl em of whether ther e ar e vertic es in a given gr aph G b elonging to c or e ( G ) is NP -har d. It has been noticed in [1] that if  is a her editary (i.e., induced subgraph clo s ed) family of graphs for which co mputing the indep endence num ber α ( G ) is poly- nomial, then core( G ) c an b e computed efficiently for G ∈  . F o r instance, it is true in the case o f p er fect gr aphs, line gra phs, circula r gr aphs, and circular arc g raphs. A sketc h o f a sequential alg o rithm computing core ( G ) for K¨ onig- Egerv´ ar y g r aphs has b een presented in [5]. In this pap er, we provide b oth sequential and par allel a lgorithms finding core( G ) in p olyno mial time, where G is a K ¨ onig-Eger v´ ary gr aph. 2 Results and Algor ithms The following result plays a key role in building our alg orithms. Theorem 5. L et G = ( V , E ) b e a K¨ onig-Egerv´ ary gr aph of or der n , and v ∈ V . 1. If µ ( G ) = µ ( G − v ) , then G − v is a K ¨ onig-Egerv´ ary gr aph and v ∈ core( G ) . 2. If µ ( G ) = µ ( G − v ) + 1 , t hen G − v is a K¨ onig-Egerv´ ary gr aph if and only if v / ∈ core( G ) . Pr o of. By definition of core : v ∈ core( G ) if a nd only if α ( G ) = α ( G − v ) + 1 . Clearly α ( G − v ) + µ ( G − v ) ≤ n − 1, α ( G ) − 1 ≤ α ( G − v ) ≤ α ( G ) and µ ( G ) − 1 ≤ µ ( G − v ) ≤ µ ( G ) ho ld for every v ∈ V ( G ). Case 1. µ ( G ) = µ ( G − v ). Assume, to the co n tra ry , that G − v is not a K¨ onig-Eger v´ ary gr aph. Hence G − v satisfies the inequality α ( G − v ) + µ ( G − v ) < n − 1 = α ( G ) + µ ( G ) − 1 , which lea ds to the follo wing contradiction: α ( G − v ) < α ( G ) − 1 . Therefore , G − v is a K¨ o nig-Ege r v´ ary graph, and, mor eov er, w e infer that α ( G − v ) = α ( G ) − 1, i.e., v ∈ cor e( G ). Case 2. µ ( G ) = µ ( G − v ) + 1. Then G − v is a K¨ onig-Eg erv´ ary g raph if and only if α ( G − v ) + µ ( G − v ) = n − 1 = α ( G ) + µ ( G ) − 1 ⇐ ⇒ α ( G ) = α ( G − v ) , i.e., v / ∈ co re( G ), and this completes the pro of. 4 Levit and Mandrescu T aking into account that every subgraph of a bipartite gra ph is bipartite, one ca n see that Theor em 5 is spe c ified as follows. Corollary 2. L et G = ( V , E ) b e a bip artite gr aph and v ∈ V . Then v ∈ core( G ) if and only if µ ( G ) = µ ( G − v ) . Let us notice that if G is a K¨ onig-Eg erv´ ary graph and has a perfect matc hing, then µ ( G ) = µ ( G − v ) + 1 ho lds for every v ∈ V ( G ). Hence by Theorem 5 we deduce the following. Corollary 3. L et G = ( V , E ) b e a K¨ onig-Egerv´ ary gr aph with a p erfe ct matching and v ∈ V . Then v ∈ core( G ) if and only if G − v is not a K¨ onig-Egerv´ ary gr ap h. Theorem 5 motiv a tes the subsequent algo rithm finding core( G ) for a ge neral K¨ onig-Eger v´ ary gr aph G . Algorithm 6 Input = a K¨ onig-Egerv´ ary gr aph G = ( V , E ) Output = core( G ) = S c ( v )=1 { v } , wher e c ( v ) =  1 , if v ∈ cor e( G ) 0 , if v / ∈ co re( G ) Se quential Complexity = O ( m • √ n ) + O ( n ∗  m • √ n + ( m + n )  ) = O ( m • n • √ n ) Par al lel Complexity with n pr o c essors = O ( m • √ n ) + O ( m • √ n + ( m + n )) = O ( m • √ n ) 1. c ompute µ ( G ) 2. for al l v ∈ V do i n p ar al lel 3. c ompute µ ( G − v ) 4. if µ ( G ) = µ ( G − v ) 5. then c ( v ) := 1 6. else c ompute k e ( v ) := G − v is a K¨ onig-Egerv´ ary gr aph 7. c ( v ) := k e ( v ) 8. co re( G ) := S c ( v )=1 { v } F or ins ta nce, a pplying Algorithm 6 for the graph G 1 from Figure 3, we get the following: – µ ( G 1 ) = 3 – µ ( G 1 − v i ) = 3 = µ ( G 1 ) for i ∈ { 6 , 7 } – µ ( G 1 − v i ) = 2 < µ ( G 1 ) for i ∈ { 1 , ..., 5 } – c ( v i ) := 1 for i ∈ { 6 , 7 } – k e ( v i ) = 1, i.e., G 1 − v i is a K¨ onig- Egerv´ ar y gra ph for i ∈ { 1 , 2 , 3 , 4 } Core of a K¨ onig-Egerv´ ary Graph 5 ✇ ✇ ✇ ✇ ✇ ✇ ✇    v 1 v 2 v 3 v 4 v 5 v 6 v 7 G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇ x 1 x 2 x 3 x 4 x 5 x 6 x 7 G 2 Fig. 3 . G 1 , G 2 are K¨ onig- Egerv´ ar y gra phs without a p erfect matching. – k e ( v i ) = 0, i.e., G 1 − v i is not a K¨ onig-Ege r v´ ary gr aph, for i ∈ { 5 } – c ( v i ) = 0 for i ∈ { 1 , ..., 4 } – c ( v 5 ) := k e ( v 5 ) = 1 – consequently , core( G 1 ) = { v 5 , v 6 , v 7 } . Prop ositio n 2. Algorithm 6 c orr e ctly c omputes c ore( G ) of a K¨ onig-Egerv´ ary gr aph G on n vertic es and m e dges, with (i) se quential time c omplexity O ( m • n • √ n ) ; (ii) p ar al lel time c omplexity with n pr o c essors O ( m • √ n ) . Pr o of. Accor ding to Theorem 5, to decide whether a vertex v ∈ V ( G ) b elongs or not to cor e( G ), one has: 1. to c ompute µ ( G ), and this requires O ( m • √ n ) time, by Theor em 2; 2. to compute µ ( G − v ) and a maximum matching M of G − v , which can b e per formed in O ( m • √ n ) time, a ccording to Theore m 2; 3. to chec k whether G − v that has M as a maxim um matching, is a K ¨ onig- Egerv´ ar y graph or no t, a nd this test can be done in O ( m + n ) time, in accorda nce with Theorem 1. Consequently , the s equential time co mplexity of Algorithm 6 is O ( m • √ n ) + O ( n ∗  m • √ n + ( m + n )  ) = O ( m • n • √ n ) , while its parallel time complexity with n pro cess o rs is O ( m • √ n ) + O ( m • √ n + ( m + n )) = O ( m • √ n ) , as claimed. If the input g r aph G is bipartite, then k e ( v ) = 1, for every v ∈ V ( G ), bec ause G − v is always bipartite, hence a K¨ onig-Ege rv´ ary graph. Co ns equently , for bipa rtite graphs we obta in the following simpler algor ithm. Algorithm 7 Input = a bip artite gr aph G = ( V , E ) Output = core( G ) = S c ( v )=1 { v } , wher e c ( v ) =  1 , if v ∈ cor e( G ) 0 , if v / ∈ co re( G ) Se quential Complexity = O ( m • √ n ) + O ( n ∗  m • √ n  ) = O ( m • n • √ n ) Par al lel Complexity with n pr o c essors = O ( m • √ n ) + O ( m • √ n ) = O ( m • √ n ) 6 Levit and Mandrescu 1. c ompute µ ( G ) 2. for al l v ∈ V do i n p ar al lel 3. c ompute µ ( G − v ) 4. if µ ( G ) = µ ( G − v ) 5. then c ( v ) := 1 6. else c ( v ) := 0 7. co re( G ) := S c ( v )=1 { v } F or example, a pplying Algorithm 7 to the gra ph G 2 depicted in Figure 3, we obtain the following: – µ ( G 2 ) = 3 – µ ( G 2 − x i ) = 2 < µ ( G 2 ) for i ∈ { 1 , ..., 5 } – µ ( G 2 − x i ) = 3 = µ ( G 2 ) for i ∈ { 6 , 7 } – c ( x i ) = 0 for i ∈ { 1 , ..., 5 } – c ( x i ) = 1 for i ∈ { 6 , 7 } – consequently , core( G 2 ) = { x 6 , x 7 } . Let us notice that, unlike bipartite gra phs, a K¨ onig-E g erv´ ary graph G with a per fect matc hing can ha ve core( G ) 6 = ∅ ; e.g., the graphs H 1 and H 2 from Figure 4 hav e at least one perfect matching and core( H 1 ) = { x } , while cor e( H 2 ) = { u , v } . ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ x H 1 ✇ ✇ ✇ ✇ ✇ ✇    u v H 2 Fig. 4 . H 1 and H 2 are K¨ onig- Egerv´ ar y gra phs with p er fect ma tc hings. If G is a K¨ onig- E gerv´ ary graph having a pe r fect matching, then clear ly , µ ( G ) = µ ( G − v ) + 1 holds for every v ∈ V ( G ). Hence, v ∈ co re( G ) if a nd only if G − v is not a K¨ onig-Eg e rv´ ary graph. Consequen tly , core( G ) of a K¨ onig-Ege rv´ ary graph G owning a p erfect matching, may be fo und mor e efficiently . Algorithm 8 Input = a K¨ onig-Egerv´ ary gr aph G with a p erfe ct matching Output = core( G ) = S c ( v )=1 { v } , wher e c ( v ) =  1 , if v ∈ cor e( G ) 0 , if v / ∈ co re( G ) Se quential Complexity = O ( n • ( m • √ n )) = O ( m • n • √ n ) Par al lel Complexity with n pr o c essors = O ( m • √ n ) 1. for al l v ∈ V ( G ) do in p ar al lel 2. c ompute k e ( v ) := G − v is a K¨ onig-Egerv´ ary gr aph 3. c ( v ) := k e ( v ) 4. co re( G ) := S c ( v )=1 { v } Core of a K¨ onig-Egerv´ ary Graph 7 Applying Algo rithm 8 for the graph G 1 from Figure 5, we get that: k e ( v 1 ) = 0 and k e ( v 3 ) = 0, i.e., G 1 − v 1 and G 1 − v 3 are not K¨ onig-Eg erv´ ary gra phs, while k e ( v 2 ) = k e ( v 4 ) = k e ( v 5 ) = k e ( v 6 ) = 1 , i.e., G 1 − v i , i ∈ { 2 , 4 , 5 , 6 } , are still K¨ onig-Eger v´ ary graphs. Cons equently , it follows that only c ( v 1 ) = c ( v 3 ) = 1 , and hence co re( G 1 ) = { v 1 , v 3 } . ✇ ✇ ✇ ✇ ✇ ✇    v 1 v 2 v 3 v 4 v 5 v 6 G 1 ✇ ✇ ✇ ✇ ✇ ✇ G 2 Fig. 5. G 1 and G 2 are K¨ onig-Eger v´ ary gr aphs with p erfect matchings. It is worth men tioning that if the input gra ph G is bipa rtite having a per- fect matching, then Algorithm 8 gives a constructive pro of of P rop osition 1 (ii) claiming that core( G ) = ∅ . F o r example, using Algorithm 8 for the bipar tite graph G 2 from Fig ure 5, one can see that k e ( v ) = 1 holds for every v ∈ V ( G 2 ), and hence, co re( G 2 ) = ∅ . 3 Conclusions In this pap er w e present a sequen tial algorithm with time complexity O ( m • n 3 2 ) finding core( G ) o f a K¨ onig-Eger v´ ary graph. Its par allel co unterpart s olves the same pr oblem in O ( m • n 1 2 ) time co mplexit y . It is known that the unique maximum independent set pr oblem is NP - ha rd for gene r al graphs [2 2]. One of a pplications of our re s ults is a p olynomia l algo - rithm recog nizing a K¨ onig-E gerv´ ary g raph with a unique maximum independent set. In fact, the graph G has a unique ma ximum independent set if and o nly if core( G ) is a maximal indep endent set [8]. Therefor e, whenever there is a p oly - nomial algorithm returning core( G ), one ca n decide in p olynomia l time whether G has a unique maximum independent set. Consequently , to reco gnize a K¨ onig- Egerv´ ar y graph G with a unique maximum indep endent set, it is eno ugh to run Algorithm 6, and then to try enla rging its output to an indep endent s et. The enlarging par t is handled in O ( m ) time complexit y sequen tially , while in par allel it ma y b e implemen ted with O (1 ) time complexity . References 1. Boros, B., Golum bic, M. C., Levit, V. E.: On the num b er of vertic es b elonging to al l maximum stable sets of a gr aph , Discrete Applied Mathematics 124 (2002) 17-25. 2. Bourjolly , J. M., Hammer, P . 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