J Braz Comput Soc (2012) 18:153–165 DOI 10.1007/s13173-011-0054-2

ORIGINAL PAPER

Contributions of Jayme Luiz Szwarcfiter to and computer science

Cláudio Leonardo Lucchesi

Received: 19 December 2011 / Accepted: 23 December 2011 / Published online: 25 January 2012 © The Brazilian Computer Society 2012

Abstract This is an account of Jayme’s contributions to followed by the affiliation, and I quote: Universidade Fed- graph theory and computer science. Due to restrictions in eral do Rio de Janeiro, Argentina. length, it is not possible to provide an in-depth coverage of I must say that the elegance and simplicity of the pa- every aspect of Jayme’s extensive scientific activities. Thus, per was worth the irritation. There was also the added plea- I describe in detail only some of his principal contributions, sure of seeing a Brazilian publishing a paper with Knuth. As touch upon some, and merely list the other articles. you know, Knuth is one of the most illustrious researchers I found it easier to write the article in the first person, as in computer science. Almost 20 years later, in 1992, Knuth though it is an account of a previously given lecture. wrote a book entitled Literate Programming [29]. Chapter 3 of this book contains a reproduction of the original article. Keywords Graph theory · Computer science · Computer This paper was one of Jayme’s first contributions to make science in Latin America Brazil better known abroad. Since then, Jayme has pro- duced a significant number of papers, some of which, like the paper mentioned above, are milestones in the history of 1 Mucho Gusto en Conocerlo Brazilian science. Jayme’s activity has also helped to fos- ter the integration of the scientific community in our coun- According to Marisa Gutierrez, this is the way people say try. He also has an important role in the integration of the “nice to meet you” in Argentina. Latin American scientific communities, through collabora- The first time I heard of Jayme was back in 1974. At that tion with many researchers from different countries. And time, I was a graduate student at the University of Waterloo, that includes colleagues from Argentina, of course. in Canada. In those days, I used to become quite upset with the general lack of knowledge about Brazil. The standard Exercise 1 Determine the list of Latin American researchers joke among the Brazilian students was that every foreigner who are coauthors of Knuth. thought that Buenos Aires was the capital of our country. A friend of mine came by my office and showed me a copy of one of Jayme’s articles, a joint paper with 2 A Universidade do Brasil Knuth [30]; a very nice paper. However, my friend’s inten- tion was not to contribute to my education, but to irritate me As many of you know, the Universidade Federal do Rio de with the title page of the article. It showed Jayme’s name, Janeiro is actually the Universidade do Brasil. Jayme par- ticipated in the creation of the three departments in that university where most of the research in computer science The author is currently at Facom-UFMS, MS, Brazil, with email address [email protected]. and graph theory takes place: (i) the Núcleo de Computação  Eletrônica, created in 1970, (ii) the Department of Com- C.L. Lucchesi ( ) puter Science, created in 1971, part of the Institute of Math- Institute of Computing, UNICAMP, C. P. 6176, 13083-970 Campinas, SP, Brazil ematics, and (iii) the Systems Engineering and Computer e-mail: [email protected] Science Program, created in 1971. 154 J Braz Comput Soc (2012) 18:153–165

Jayme taught the first computer science undergraduate A topological sorting of a directed graph G is an enumer- course at the Universidade do Brasil, “Introduction to Com- ation T := (v1,v2,...,vn) of the n vertices of G such that puting,” to undergraduate Physics students, back in 1971. for each edge (vi,vj ) of G, i

Let G∞ denote the infinite graph whose vertices are the Hamiltonian path has its origin and terminus in distinct parts points in the plane having integral coordinates and in which of the bipartition. two vertices are adjacent if and only if the Euclidean dis- Without loss of generality, let us assume that G = tance between them is equal to one. R(m,n), m ≥ n. A necessary condition for (G,s,t) to have ∞ A grid graph is a finite vertex- of G . a solution is that the parities of s and t coincide if and only Thus, a grid graph is completely specified by its set of ver- if (i) mn is odd and (ii) each of s and t is even. The authors tices. For each vertex v,letvx and vy denote the coordinates call this the color compatibility condition of the (G,s,t) + of v.Theparity of v is the parity of the sum vx vy of its problem. + coordinates. Thus, v is even if vx vy is even, and is odd The color compatibility condition is not sufficient in gen- otherwise. All grid graphs are bipartite, with the edges con- eral. It is sufficient for “large” rectangles, in which both di- nectinganevenvertextoanoddvertex. mensions are at least four. It is also sufficient for rectan- Denote by R(m,n) the grid graph whose set of vertices gles in which m is odd and n = 3 (under the hypothesis that { : ≤ ≤ ≤ ≤ } is v 1 vx m, 1 vy n .Arectangular graph G is a m ≥ n). grid graph that is isomorphic to R(m,n), for some integers The problems appear when either n ≤ 2 or when n = 3 m and n, called the dimensions of G. Note that the dimen- and m is even. If n = 1, then the problem has a solution if sions of a rectangular graph completely specify the graph, and only if up to isomorphism. Let s and t be distinct vertices of a grid graph G.The {s ,t }={1,m}. (1) Hamilton path problem (G,s,t) consists in determining x x whether there is in G a Hamiltonian path from s to t. = The problem of determining whether or not an instance If n 2, then the problem has a solution if and only if (G,s,t) of a Hamilton path problem has a solution is NP- = = ∈{ } complete. This result was proven by Jayme and his coau- sx tx or sx tx 1,m . (2) thors in the paper I just mentioned. This result is the most important known NP-complete restriction of the Traveling The description of the necessary and sufficient condition for Salesman Problem. Indeed, the book The Traveling Sales- the case in which n = 3 and m is even is more elaborate: We man Problem [31], edited by Lawler et al., contains a chap- have seen that the color compatibility condition implies that ter written by Johnson and Papadimitriou where the authors in this case one of s and t must be even, the other odd; adjust mention the importance of the NP-completeness result and the notation so that s is odd and t is even. The additional transcribe its proof as originally published in [26]. condition is then the following. The proof of the NP-completeness is too technical to be Under the hypothesis that m is even, n = 3, s is odd and presented here. But it uses grid graphs G that have “holes,” t is even, the instance (G,s,t)has a solution if and only if that is, G∞ − G is not connected. In the paper, the authors indicate that it is not known whether or not the Hamilton sx ≥ tx − 1, with equality only if ty = 2. (3) path problem is NP-complete for grid graphs without holes. The paper also presents a very nice positive result In sum, the color compatibility condition is necessary. As- for rectangular graphs: for every rectangular graph G := sume that m ≥ n. The condition is sufficient for the cases in R(m,n), there is an algorithm of complexity O(mn) that which n ≥ 4. It is also sufficient for the case in which n = 3 either determines that the instance (G,s,t) of the Hamilton and m is odd. For the cases in which (i) n = 1, or (ii) n = 2, path problem has no solution or determines a solution. As or (iii) n = 3 and m is even, the condition has the additional usual, the algorithm is a by-product of the proof of theo- requirement of (1), (2), and (3), respectively. rems. I will not present the details here, but let me at least As I said, there are too many details to describe the proof tell you the characterization of the instances that do have a and the algorithm here. The following exercises may prompt solution. you to try to prove the result. Consider a rectangular graph G of dimensions m and n. We have seen that G is bipartite. If m and n are both odd,  Exercise 3 Assume that sx,tx ≤ m − 2. Let G := R(m − then the number mn of the vertices of G is odd, one of the  parts of the bipartition has one vertex more than the other 2,n). Show that if the problem (G ,s,t)has a solution then so too does (G,s,t). Prove also that if n = 3 and (G,s,t) part. In that case, every Hamiltonian path in G must start  and end in vertices that lie in the majoritarian part. On the has a solution then (G ,s,t)has a solution. other hand, if at least one of m and n is even, then G has an even number of vertices, and each part of the bipartition of G Exercise 4 Prove that the additional conditions (1), (2), has precisely half of the vertices of G. In that case, every and (3) are necessary for small rectangles. J Braz Comput Soc (2012) 18:153–165 157

7 Optimal multiway search trees key is equal to h(x(ei)) if y = yi for some key ei ; otherwise, the cost is h(gi) − 1, where y ∈ Ii . As you see, the cost is The most popular data structure used in large databases is the number of input operations from disk that are necessary the B-tree, or a variation thereof. B-trees were invented by in order to have the answer to the problem “does there exist Bayer and McCreight in 1971 [5], in order to minimize the akeyei such that y = yi ?” Taking into account the proba- number of input operations in secondary storage. In fact, in bilities mentioned earlier, we deduce that the average search the worst case, this number is the height of the tree, and cost of T , or simply, the cost of T ,isthesum a B-tree may hold millions of keys with a height equal to n n three. This means that in order to find a key in a database of     pih x(ei) + qi h(gi) − 1 . that size it suffices to perform at most three input operations i=1 i=0 from disk. I am now going to describe a solution found by Jayme One of the problems is to minimize the cost of a tree, to an open problem posed by McCreight himself. That solu- given the limit L, in the generic case of multiway trees, or tion was published in 1984 [43]. I shall also describe some the limits (L1,L2), in the case of a weak B-tree. related work published in that same paper. For this, we need There is a particular case of the problem that is solvable some definitions. in polynomial time, using standard dynamic programming Let E := (e1,...,en) be a sequence of elements called techniques. It is the case in which the sizes of the keys are keys. Associated with each key ei there is its size si and its equal (see [22, 25]). value, an integer yi . We assume that y1

The n + 1leavesofT are called gaps and denoted g0,g1, He also proved another similar result in that paper, which ...,gn. Each gap gi corresponds to the interval Ii := {y : implies the previous one. yi

M keys in the root and minimum root size. As usual, we L2, then A[n + 1,M + 1] is the size of the root of a weak ≤ are assuming that 0 L1, otherwise there is no reason for a tree of height in the root and minimum root size. A straightforward imple- two. mentation of this algorithm takes time O(n2M). Jayme was Let us first show how Jayme solved a simpler problem: able to lower the complexity to O(nM log n). solve McCreight’s problem relaxing the condition on the number of keys in the root. That is, the root may have any number of keys, subject to the upper limit condition on its 8 Iterated clique graphs with increasing diameters size. Here is a description of the very elegant solution for this Let us now examine a result proved by Jayme and Born- simpler problem. Define a complete, acyclic directed graph stein [12]. The result answered a question that had been open D on the set {v0,v1,...,vn+1} of vertices, in which for ev- for 12 years, it was posed in two articles and also in the book ery pair (i, j) such that 0 ≤ i0. Likewise, for optimal) solution to the simplified problem: the set of keys each nonnegative integer i, define Li(G) := G if i = 0 and − in the root is precisely the set {ei : vi ∈ V(P)− v0 − vn+1} Li(G) := L(Li 1(G)),ifi>0, where L(G) denotes the and the size of the root is the cost of P . If, in addition, the line graph of G. For each vertex k of K(G), and each posi- −i cost of P is minimum, then the tree has minimum root size. tive integer i,theith inverse image K (k) of k is the clique = −(i−1) Thus, Jayme reduced the simplified version of the prob- k of G if i 1, otherwise it is v∈K−1(k) K (v). lem to that of finding in D a path of minimum cost from v0 The diameter diam(G) of G is the maximum distance to vn+1. A straightforward implementation of a minimum between any two vertices of G.Thatis,ifd(v,w) de- cost path algorithm in D takes time O(n2). Well, Jayme dis- notes the distance between v and w in G, then diam(G) := covered a structure in the costs in the graph that allowed him max{d(v,w) : v,w ∈ V(G)}. A pair of vertices of G is dia- to decrease the complexity of the algorithm to O(nlog n). metrical if their distance is equal to the diameter of G. Let us now see how to solve the original problem, in Hedman [24] showed that which we would like to have not only minimum root size,   but also precisely M keys in the root. The solution described diam(G) − 1 ≤ diam K(G) ≤ diam(G) + 1. here seems simpler than that originally given by Jayme in the paper, but is somewhat equivalent. In terms of the com- The proof of the above inequalities is not difficult. In fact, if plete acyclic directed graph D defined above, the problem is you want to warm up for this subject, here is a nice problem. equivalent to that of finding in D, among all directed paths Exercise 5 Let G be a connected graph with at least one of length M + 1 from v0 to vn+1, one that has minimum cost. edge. Let F be a family of nonnull cliques of G such that So, it is easy to solve: define an (n + 1) × (n + 1) matrix for every edge e of G, both ends of e lie in some member of A, initialized with ∞ everywhere, except at entry A[0, 0], F .LetI denote the intersection graph of G induced by F , which is initialized with zero. Then that is the set of vertices of I is F , two vertices f1 and f2 of I are adjacent if and only if the cliques f1 and f2 of G for r = 1,...,M + 1 intersect. Prove that for i = 0,...,n diam(G) − 1 ≤ diam(I) ≤ diam(G) + 1. for j = i + 1,...,n+ 1   let A[j,r]:=min A[j,r],A[i, r − 1]+d[i, j] . Prove also that diam(I) = diam(G) + 1 if and only if there exist two cliques k1 and k2 in F such that for each vertex [ ] Of course, A i, r is the cost of the minimum cost path from v1 of k1 and each vertex v2 of k2, vertices v1 and v2 are v0 to vi that has length r. In particular, if A[n + 1,M+ 1]≤ diametrical in G. J Braz Comput Soc (2012) 18:153–165 159

Note that the inequality proved by Hedman is a particular In that paper, the authors characterize the clique graphs case of Exercise 5. It suffices to define F to be the family of of two families of graphs, the directed path graphs, and the maximal cliques of G: graph I will be the clique graph of rooted path graphs. G in that case. Another application of Exercise 5 is to define A directed path graph (or a DV graph) is the intersection F to be set of pairs of vertices of G that are adjacent: in that graph of the family of directed paths of a directed tree. A du- case, graph I will be the line graph of G. The inequality for ally directed path graph (or dually DV graph) is a graph G line graphs was proved by Knor et al. [28]. that admits a spanning directed tree T such that, for each Hedman also described a family of graphs G for which edge (v, w) of G, T contains a directed v − w path or a diam(K(G)) = diam(G) + 1, and asked if graphs G exist directed w − v path whose vertices form a clique in G. with diam(Ki(G)) = diam(G) + i for each positive integer In order to describe their characterization of a dually DV i ≥ 2. graph, we need some definitions. The existence of such graphs G for i = 2 was estab- A family F of sets has the Helly property if, for every lished by Balakrishnan and Paulraja [2] and independently nonnull subcolletion G of F, either G contains two disjoint by Peyrat, Rall, and Slater [34], who also proved the exis- sets or all the sets in G have a common element. A graph is tence in the cases i = 3 and i = 4. clique-Helly if its family of maximal cliques has the Helly property. Jayme and Claudson were able to prove the existence of  such a G for all positive integers i. In order to do that, they For any graph G,letG denote the graph obtained from G by adding, for each vertex v of G, a new vertex v and a defined a graph H(d,G), where d is a positive integer and  G is a graph. Graph H(d,G)was defined as follows: (i) take new edge joining v to v . two copies G and G of G; (ii) take a new vertex v, and join Here is the characterization of dually DV graphs. it to each vertex of each of G and G by a path of length d. Then they showed the following result. Theorem 7 A graph G is a dually DV graph if and only if G is clique-Helly and K(G) is a DV graph. Theorem 5 Let G be a graph whose diameter is at most The authors also derive an algorithm of complexity 2d, v and v two vertices of Ki(H (d, G)). If the inverse 1 2 O(|E(G)|4) to determine whether a given graph G is du- ith images of v and v are contained in G and G, re- 1 2 ally DV. spectively, then v and v are diametrical with distance 1 2 Let us now describe their characterization of clique diam(H (d, G)) + i. graphs of rooted path graphs. A rooted tree is a directed tree having precisely one vertex with in-degree zero (a rooted Thus, Hedman’s problem was reduced to finding a suit- tree is sometimes called a branching). A rooted path graph able graph G. They were able to use a relatively compli- (or RDV graph) is the intersection graph of the family of i cated argument to show that L (Kn) is a good choice. More directed paths of a rooted tree. A dually rooted path graph specifically, they proved the following result. (or dually RDV graph) is a graph G that admits a spanning rooted tree T such that, for each edge (v, w) of G, T con- Theorem 6 Let i, d and n be positive integers such that tains a directed v − w path or a directed w − v path whose 2d>i+ 2, n ≥ 4 and i ≤ n/2 −1. Let G be the graph vertices form a clique in G. i H(d,L (Kn)). Then A is a graph that contains no induced cycle   of length four or greater. A strong chord of a cycle of a graph i+1 = + + diam K (G) diam(G) i 1. is a chord that joins two vertices of the cycle with an odd distance in the cycle. A strongly chordal graph is a graph in With the above theorem, they solved completely the which every cycle on six or more vertices contains a strong question, which had remained open for 12 years. chord. Here is the characterization for dually RDV graphs.

9 Clique graphs of directed path graphs and of rooted Theorem 8 The following statements are equivalent for a path graphs graph G: (i) G is a dually RDV graph. Let us now examine another important contribution of (ii) G is clique-Helly and K(G) is an RDV graph. Jayme. It is a joint work with Prisner [36]. The paper was (iii) G is strongly chordal and K(G) is an RDV graph. selected to be part of the Editors’ Choice 1999 of the Dis- crete Applied Mathematics, an indication that it was one the The authors also derive an algorithm O(|V(G)2.38|) to best articles published by the periodical in 1999. determine whether or not a given graph G is dually RDV. 160 J Braz Comput Soc (2012) 18:153–165

10 Other results pair of vertices in S is an even pair and each vertex of S is a source of some transitive orientation. We now examine very briefly other results obtained by In the second paper of the series, a new author joins the Jayme. team: Gimbel [19]. The authors find a condition that is nec- essary and sufficient for the problem to have a solution. For 10.1 Comparability graphs a specified set S of sources and a specified set T of sinks, the authors construct a graph G(S, T ) that is trivially ob- Let D be an acyclic orientation of a graph G. Then D is tained from G, S, and T and has size linear on the size of transitive if, for each pair (u, v) and (v, w) of edges of D, G. Then they show that the problem has a solution if and edge (u, w) also lies in D. A graph is a only if G(S, T ) is a comparability graph. So, not only they if it admits a transitive orientation. solve the problem from a mathematical point of view, but For any two vertices v and w of D,let v,w denote the they also give a polynomial algorithm for deciding whether set consisting of those vertices that are simultaneously de- the problem has a solution. scendants of v and ancestors of w. Orientation D is locally Finally, in the third paper of the series [20], they char- transitive if G[ v,w ] is transitive, for each edge (v, w) of acterize even and odd pairs in comparability and in P4- D. Graph G is local comparability if it admits a locally tran- comparability graphs. The characterizations lead to sim- sitive orientation. ple algorithms for deciding whether a given pair of ver- tices forms an even or odd pair in these classes of graphs. A graph G is P4-comparability if it admits an orienta- + tion D such that the restriction of D to the subgraph of G The complexities of the proposed algorithms are O(n m) 2 spanned by the set of vertices of each path of length three is for comparability graphs and O(n m) for P4-comparability transitive. graphs. The former represents an improvement over a recent A is the intersection graph of chords of a algorithm of complexity O(nm). circle, in which no two chords have a common point in the circle. 10.2 Cliques A pair {v,w} of vertices of G is even if every induced There is an enormous number of significant results involving path from v to w has even length. A pair {v,w} is odd if v cliques. Some of these have already been described. Here are and w are nonadjacent and each induced path from v to w some more. has odd length. There are four papers on this subject that should be men- 10.2.1 Clique graphs free of K and K tioned. In the first one [48], Jayme introduces the concept 3 4 of local comparability graphs, as a generalization of com- This is joint work with Protti [37]. The authors characterize parability graphs. The class of local comparability graphs the graphs whose clique graphs are free of triangles in terms includes the comparability graphs and the circle graphs. of forbidden induced subgraphs: K1,3, the 4-fan and K4.The The first main result in that paper is that every local com- 4-fan is the graph obtained from the 4-wheel by deleting an parability graph is a difference of two comparability graphs. edge from the rim. They give a similar characterization for The second main result is that the class of local comparabil- graphs whose clique graphs are free of K4. ity graphs of dimension 1 is precisely the class of connected interval graphs that correspond to a set of totally noncom- 10.2.2 Clique-inverse graphs of bipartite graphs parable intervals of the real line. Circle graphs are similarly but less concisely characterized. This is also joint work with Protti [39]. The authors charac- The next three papers show a beautiful evolution of terize the families of graphs whose clique graphs are bipar- thought, culminating with a nice characterization of source tite, in terms of forbidden configurations. The clique graph and sink sets, and also a characterization of even and odd of a graph G is bipartite if and only if G is free of induced pairs in a comparability graph. subgraphs in the following list: K1,3, the 4-fan, the 4-wheel, The first of these three papers considers the problem of C2n+5 (n ≥ 0). They also characterize two more classes: determining whether a comparability graph has a transitive (i) those graphs whose clique graphs are chordal bipartite orientation with specified sources and sinks. It is a joint graphs and (ii) those graphs whose clique graphs are a tree. work with Mello and Figueiredo [58]. They consider clique partitions of a comparability graph and determine some nec- 10.2.3 Clique graphs with linear size essary conditions for the existence of a solution to the prob- lem. This condition turns out to be sufficient for graphs with Another joint work with Protti [38]. Let G be a graph. By ex- at most three maximal cliques. In particular, if only sources amining K(G), the authors describe some sufficient condi- are specified, then the set is a source set if and only if each tions for the number of maximal cliques of G to be bounded J Braz Comput Soc (2012) 18:153–165 161 by O(|V(G)|). These conditions are then applied to analyze 10.2.8 Computing all maximal cliques distributedly the complexity of recognizing clique-inverse graphs of var- ious classes of graphs. In some cases, polynomial time al- This is joint work with Protti and França [40]. The au- −1 gorithms are obtained, such as in the case of K (Kr -free). thors present a parallel algorithm for generating all maxi- In other cases, the bound is used to show that certificates mal cliques of a graph. The of the algo- may be verified in polynomial time, within a proof of NP- rithm is restricted to the induced neighborhood of a vertex completeness. and the communication complexity is O(MΔ), where M is the number of connections and Δ the maximum degree in 10.2.4 Clique-Helly graphs the graph.

In this paper [49], Jayme describes a characterization of 10.2.9 Enumeration of maximal cliques of a circle graph clique-Helly graphs, leading to a polynomial time algorithm for recognizing them. This is joint work with Barroso [51]. The authors apply the notion of locally edge transitive orientations of an undi- 10.2.5 Clique-complete graphs rected graph and obtain an algorithm for generating all max- imal cliques of a circle graph G in time O(n(m+α)), where This is a joint work with Lucchesi and Mello [32]. At the n, m, and α are the number of vertices, edges and maximal time, Mello had just completed her doctoral thesis, under cliques of G. In addition, they show that the actual number the supervision of Jayme. Some years prior to that, Mello of such cliques can be computed in O(nm) time. had written her Master’s dissertation under my supervision. 10.2.10 Maximal cliques in circle graphs So, it was a very pleasant opportunity to be a coauthor with Jayme and a former student of both of us. This is joint work with Cáceres and Song [14]. A Coarse For a natural number n, a graph G is n-convergent if Grained Multicomputer (CGM) consists of a set of p pro- Kn(G) is isomorphic to K , the one-vertex graph. A graph 1 cessors with O(N/p) local memory per processor and an G is convergent if it is n-convergent for some natural arbitrary communication network (or a shared memory). number n. A 2-convergent graph is called clique-complete. A CGM algorithm consists of alternating local computation A universal vertex is a vertex adjacent to every vertex of the and global communication rounds. At each communication graph. round, each processor sends and receives O(N/p) data. The authors describe the family of minimal graphs which In this paper, the authors present a parallel algorithm for are clique-complete but have no universal vertices. The min- finding the maximal cliques of a circle graph using the CGM imality used there refers to induced subgraphs. In addition, model. The proposed algorithm requires O(log p) commu- they show that recognizing clique-complete graphs is Co-NP nication rounds. In a regular, sequential depth search, nor- complete. mally each edge is visited a constant number of times. The authors devised a new technique, called the unrestricted 10.2.6 Clique convergent graphs depth search, in which each edge may be visited an un- bounded (but finite) number of times. The authors regard This is a joint work with Bornstein [11]. The index of a this technique as the main contribution of the paper. The convergent graph G is the smallest n such that G is n- three authors also have another paper on unrestricted depth convergent, while its Helly defect is the smallest n such that search in parallel [15]. Kn(G) is clique-Helly. Bandelt and Prisner [3] proved that the Helly defect of a chordal graph is at most one and asked 10.3 Edge clique graphs whether there is a graph whose Helly defect exceeds the dif- ference of its index and diameter by more than one. In this The edge clique graph Ke(G) of a graph G is the graph paper, an affirmative constructive answer to the above ques- whose set of vertices is the set of edges of G, two vertices of ≥ tion is given: for any arbitrary finite integer n 0 a graph is Ke(G) are adjacent if and only if the corresponding edges exhibited in which the Helly defect exceeds by n the differ- lie in a (common) clique of G. ence of its index and diameter. An edge component of a graph G is a component of its edge clique graph. 10.2.7 Clique graphs of chordal graphs and of path graphs 10.3.1 Characterization of edge clique graphs Another joint work with Bornstein [52], where the authors characterize the clique graphs of chordal graphs and the This is joint work with Cerioli [16]. A k-labeling of a clique graphs of path graphs. graph G with n vertices is an assignment of a set l(v) ⊂ 162 J Braz Comput Soc (2012) 18:153–165

{1, 2,...,n} to each vertex v of G, such that |l(v)|=k and 10.4.2 Enumeration of kernels all label  sets are distinct. A set S of vertices is triangular if | |= r S 2 for some integer r.AsetS of vertices is strongly  A kernel N of a directed graph D is an independent set of | |= |l(S)| triangular, with respect to a 2-labeling l,if S 2 . vertices of D such that for every w ∈ V(D)− N there is an The authors show that a graph G is an edge clique graph if edge from w to N. The existence of a kernel in an directed and only if it has a 2-labeling that satisfies the following two graph with no odd directed cycles was proved by Richard- properties: (i) every maximal clique is strongly triangular son [41]. and (ii) every strongly triangular set is a clique. This is a joint work with Chaty [53]. The authors give an algorithm for generating all distinct kernels in a directed 10.3.2 Starlike graphs graph D with no odd directed circuits. The complexity of the algorithm is O(nm(k + 1)), where n, m, and k are the num- Denote by N(v) the set of vertices that are adjacent to v ber of vertices, edges, and kernels of D. Also, they show that G N[v] {v}∪N(v) G in a graph and by the set . A graph the problem of determining the number of kernels in a di- is starlike if there exists a partition C,D ,...,D (s ≥ 0) 1 s rected graph D is #P -complete, even if the longest directed of the set of vertices of G such that C is a maximal clique circuit of D has length two. and, for u ∈ Di , v ∈ Dj , i = j implies that {u, v}∈E(G), whereas i = j implies that N[u]=N[v]. It follows that 10.4.3 A minimax equality each Di is included by precisely one maximal clique Ci , and Di = Ci − C. If, in addition, C ∩ Ci ⊂ C ∩ Ci+1 for 1 ≤ i

10.4.6 Rooted tree structure 10.6.2 Euler tours

A directed graph D = D(V,E) with a given root vertex s A joint work with Cáceres, Deo, and Sastry [13] describes an is reducible if every depth-first search tree with root s has alternative implementation of Atallah and Vishkin’s parallel the same set B of back edges. Thus, for a reducible directed algorithm for finding an Euler tour of a graph [1]. graph D, the associated dag (the subgraph with vertex set V and edge set E − B) is uniquely defined. A tree reducible 10.6.3 Search graph is a reducible subgraph for which the transitive reduc- tion (a smallest directed graph with the same reachability) of I should mention here three papers. The first paper is a joint the associated dag is an arborescence (outdirected tree) with work with Wilson, on ternary trees [56]. The second paper root s. is joint work with Navarro et al., on optimal binary search In this paper [44], Jayme gives the polynomial algorithm trees with costs depending on the access paths [59]. for (1) recognizing, (2) finding isomorphisms between, and The third paper is a joint work with Moscarini and Pe- (3) finding minimum equivalent directed graphs for tree re- treschi, Node Searching and Starlike Graphs.Itisaveryin- ducible graphs. teresting paper. Let G be a graph whose vertices are contam- inated. Assigning a searcher to a contaminated vertex makes it become guarded. Removing the searcher of a guarded ver- 10.5 Split-indifference graphs tex turns it clear. However, a clear vertex becomes contam- inated again if it has a contaminated neighbor. The node- This is a joint work with Ortiz and Maculan [33]. An indif- search number of G is the least number of searchers needed ference graph is an intersection graph on a set of unit in- to clear all its vertices. Gustedt [23] has shown that the prob- tervals on the real line. A split-indifference graph is a split lem of determining the node search number of G is NP-hard graph that is also an indifference graph. The authors give the for uniform k-starlike graphs. These graphs are generaliza- following characterization of split-indifference graphs. tions of split graphs, obtained when each vertex of the inde- pendent set of the bipartition of the split graph is replaced by Theorem 9 A connected graph G is split-indifference if and a k-vertex clique. The authors describe necessary and suf- only if ficient conditions for finding the node-search number of a (i) G is complete, or uniform k-starlike graph. The characterization described ex- (ii) G is the union of two cliques G1 and G2 such that G1 − tends a corresponding result for split graphs by Kloks [27]. G2 = K1, or In addition, it leads to a new algorithm for finding the node- (iii) G is the union of three cliques G1, G2, G3 such that search number for graphs of this class. G1 − G2 = K1, G2 − G3 = K1 and Acknowledgements I would like to thank Celina M.H. de Figueiredo V(G ) ∩ V(G ) =∅ or V(G ) ∪ V(G ) = V(G). and Valmir C. Barbosa for having invited me to write this article. Their 1 3 1 3 kind invitation gave me the opportunity to really grasp the extent of Jayme’s work over the years. I hope that this article will help others to Using that characterization, they determine the chromatic appreciate the breadth of Jayme’s work. index χ(G) of split-indifference graphs. In order to do that, I would also like to thank Valmir C. Barbosa, Márcia Cerioli, Celina ≥ − M.H. de Figueiredo, Sulamita Klein, and U.S.R. Murty for reading an they construct an edge coloring of K2n, n 3, using 2n 1 earlier draft and giving me many helpful suggestions. colors such that K2n has a perfect matching without color Supported by a grant from CNPq. repetitions. The resulting algorithm is very simple. It determines in linear time an optimum edge coloring of a split-indifference References graph. 1. Atallah M, Vishkin U (1984) Finding Euler tours in parallel. J Comput Syst Sci 29(3):330–337 10.6 Other results 2. Balakrishnan R, Paulraja P (1986) Self-clique graphs and diame- ters of iterated clique graphs. Util Math 29:263–268 There are many other results that I should describe, but 3. Bandelt H-J, Prisner E (1991) Clique graphs and Helly graphs. J length restrictions force me to be very concise. Comb Theory, Ser B 51(1):34–45 4. Barbosa VC, Szwarcfiter JL (1999) Generating all the acyclic ori- entations of an undirected graph. Inf Process Lett 72(1–2):71–74 10.6.1 Task scheduling 5. Bayer R, McCreight E (1971) Organization and maintenance of logic ordered indexes. Acta Inform 1:173–189 6. Bła˙zewicz J, Kubiak W, Röck H, Szwarcfiter J (1987) Minimizing Jayme has four papers in this area, three of them with mean flow-time with parallel processors and resource constraints. Bła˙zewicz and Kubiak [6–8, 45]. Acta Inform 24(5):513–524 164 J Braz Comput Soc (2012) 18:153–165

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