We now examine very briefly other results obtained by Jayme.
Comparability graphs
Let D be an acyclic orientation of a graph G. Then D is transitive if, for each pair (u,v) and (v,w) of edges of D, edge (u,w) also lies in D. A graph is a comparability graph if it admits a transitive orientation.
For any two vertices v and w of D, let 〈v,w〉 denote the set consisting of those vertices that are simultaneously descendants of v and ancestors of w. Orientation D is locally transitive if G[〈v,w〉] is transitive, for each edge (v,w) of D. Graph G is local comparability if it admits a locally transitive orientation.
A graph G is P_{4}comparability if it admits an orientation D such that the restriction of D to the subgraph of G spanned by the set of vertices of each path of length three is transitive.
A circle graph is the intersection graph of chords of a circle, in which no two chords have a common point in the circle.
A pair {v,w} of vertices of G is even if every induced path from v to w has even length. A pair {v,w} is odd if v and w are nonadjacent and each induced path from v to w has odd length.
There are four papers on this subject that should be mentioned. In the first one [48], Jayme introduces the concept of local comparability graphs, as a generalization of comparability graphs. The class of local comparability graphs includes the comparability graphs and the circle graphs.
The first main result in that paper is that every local comparability graph is a difference of two comparability graphs. The second main result is that the class of local comparability graphs of dimension 1 is precisely the class of connected interval graphs that correspond to a set of totally noncomparable intervals of the real line. Circle graphs are similarly but less concisely characterized.
The next three papers show a beautiful evolution of thought, culminating with a nice characterization of source and sink sets, and also a characterization of even and odd pairs in a comparability graph.
The first of these three papers considers the problem of determining whether a comparability graph has a transitive orientation with specified sources and sinks. It is a joint work with Mello and Figueiredo [58]. They consider clique partitions of a comparability graph and determine some necessary conditions for the existence of a solution to the problem. This condition turns out to be sufficient for graphs with at most three maximal cliques. In particular, if only sources are specified, then the set is a source set if and only if each pair of vertices in S is an even pair and each vertex of S is a source of some transitive orientation.
In the second paper of the series, a new author joins the team: Gimbel [19]. The authors find a condition that is necessary and sufficient for the problem to have a solution. For a specified set S of sources and a specified set T of sinks, the authors construct a graph G(S,T) that is trivially obtained from G, S, and T and has size linear on the size of G. Then they show that the problem has a solution if and only if G(S,T) is a comparability graph. So, not only they solve the problem from a mathematical point of view, but they also give a polynomial algorithm for deciding whether the problem has a solution.
Finally, in the third paper of the series [20], they characterize even and odd pairs in comparability and in P_{4}comparability graphs. The characterizations lead to simple algorithms for deciding whether a given pair of vertices forms an even or odd pair in these classes of graphs. The complexities of the proposed algorithms are O(n+m) for comparability graphs and O(n^{2}m) for P_{4}comparability graphs. The former represents an improvement over a recent algorithm of complexity O(nm).
Cliques
There is an enormous number of significant results involving cliques. Some of these have already been described. Here are some more.
Clique graphs free of K_{3} and K_{4}
This is joint work with Protti [37]. The authors characterize the graphs whose clique graphs are free of triangles in terms of forbidden induced subgraphs: K_{1,3}, the 4fan and K_{4}. The 4fan is the graph obtained from the 4wheel by deleting an edge from the rim. They give a similar characterization for graphs whose clique graphs are free of K_{4}.
Cliqueinverse graphs of bipartite graphs
This is also joint work with Protti [39]. The authors characterize the families of graphs whose clique graphs are bipartite, in terms of forbidden configurations. The clique graph of a graph G is bipartite if and only if G is free of induced subgraphs in the following list: K_{1,3}, the 4fan, the 4wheel, C_{2n+5} (n≥0). They also characterize two more classes: (i) those graphs whose clique graphs are chordal bipartite graphs and (ii) those graphs whose clique graphs are a tree.
Clique graphs with linear size
Another joint work with Protti [38]. Let G be a graph. By examining K(G), the authors describe some sufficient conditions for the number of maximal cliques of G to be bounded by O(V(G)). These conditions are then applied to analyze the complexity of recognizing cliqueinverse graphs of various classes of graphs. In some cases, polynomial time algorithms are obtained, such as in the case of K^{−1}(K_{
r
}free). In other cases, the bound is used to show that certificates may be verified in polynomial time, within a proof of NPcompleteness.
CliqueHelly graphs
In this paper [49], Jayme describes a characterization of cliqueHelly graphs, leading to a polynomial time algorithm for recognizing them.
Cliquecomplete graphs
This is a joint work with Lucchesi and Mello [32]. At the time, Mello had just completed her doctoral thesis, under the supervision of Jayme. Some years prior to that, Mello had written her Master’s dissertation under my supervision. So, it was a very pleasant opportunity to be a coauthor with Jayme and a former student of both of us.
For a natural number n, a graph G is nconvergent if K^{n}(G) is isomorphic to K_{1}, the onevertex graph. A graph G is convergent if it is nconvergent for some natural number n. A 2convergent graph is called cliquecomplete. A universal vertex is a vertex adjacent to every vertex of the graph.
The authors describe the family of minimal graphs which are cliquecomplete but have no universal vertices. The minimality used there refers to induced subgraphs. In addition, they show that recognizing cliquecomplete graphs is CoNP complete.
Clique convergent graphs
This is a joint work with Bornstein [11]. The index of a convergent graph G is the smallest n such that G is nconvergent, while its Helly defect is the smallest n such that K^{n}(G) is cliqueHelly. Bandelt and Prisner [3] proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the difference of its index and diameter by more than one. In this paper, an affirmative constructive answer to the above question is given: for any arbitrary finite integer n≥0 a graph is exhibited in which the Helly defect exceeds by n the difference of its index and diameter.
Clique graphs of chordal graphs and of path graphs
Another joint work with Bornstein [52], where the authors characterize the clique graphs of chordal graphs and the clique graphs of path graphs.
Computing all maximal cliques distributedly
This is joint work with Protti and França [40]. The authors present a parallel algorithm for generating all maximal cliques of a graph. The time complexity of the algorithm is restricted to the induced neighborhood of a vertex and the communication complexity is O(MΔ), where M is the number of connections and Δ the maximum degree in the graph.
Enumeration of maximal cliques of a circle graph
This is joint work with Barroso [51]. The authors apply the notion of locally edge transitive orientations of an undirected graph and obtain an algorithm for generating all maximal cliques of a circle graph G in time O(n(m+α)), where n, m, and α are the number of vertices, edges and maximal cliques of G. In addition, they show that the actual number of such cliques can be computed in O(nm) time.
Maximal cliques in circle graphs
This is joint work with Cáceres and Song [14]. A Coarse Grained Multicomputer (cgm) consists of a set of p processors with O(N/p) local memory per processor and an arbitrary communication network (or a shared memory). A cgm algorithm consists of alternating local computation and global communication rounds. At each communication round, each processor sends and receives O(N/p) data.
In this paper, the authors present a parallel algorithm for finding the maximal cliques of a circle graph using the cgm model. The proposed algorithm requires O(logp) communication rounds. In a regular, sequential depth search, normally each edge is visited a constant number of times. The authors devised a new technique, called the unrestricted depth search, in which each edge may be visited an unbounded (but finite) number of times. The authors regard this technique as the main contribution of the paper. The three authors also have another paper on unrestricted depth search in parallel [15].
Edge clique graphs
The edge clique graphK_{
e
}(G) of a graph G is the graph whose set of vertices is the set of edges of G, two vertices of K_{
e
}(G) are adjacent if and only if the corresponding edges lie in a (common) clique of G.
An edge component of a graph G is a component of its edge clique graph.
Characterization of edge clique graphs
This is joint work with Cerioli [16]. A klabeling of a graph G with n vertices is an assignment of a set l(v)⊂{1,2,…,n} to each vertex v of G, such that l(v)=k and all label sets are distinct. A set S of vertices is triangular if \(S=\binom{r}{2}\) for some integer r. A set S of vertices is strongly triangular, with respect to a 2labeling l, if \(S =\binom{l(S)}{2}\). The authors show that a graph G is an edge clique graph if and only if it has a 2labeling that satisfies the following two properties: (i) every maximal clique is strongly triangular and (ii) every strongly triangular set is a clique.
Starlike graphs
Denote by N(v) the set of vertices that are adjacent to v in a graph G and by N[v] the set {v}∪N(v). A graph G is starlike if there exists a partition C,D_{1},…,D_{
s
} (s≥0) of the set of vertices of G such that C is a maximal clique and, for u∈D_{
i
}, v∈D_{
j
}, i≠j implies that \(\{u,v\} \not \in E(G)\), whereas i=j implies that N[u]=N[v]. It follows that each D_{
i
} is included by precisely one maximal clique C_{
i
}, and D_{
i
}=C_{
i
}−C. If, in addition, C∩C_{
i
}⊂C∩C_{i+1} for 1≤i<s, then G is a starlikethreshold graph.
A generalized starlike graph is a graph G such that precisely one of its edge components is a starlike graph, the others complete graphs.
A generalized starlikethreshold graph is a graph G such that precisely one of its edge components is a starlikethreshold graph, the others complete graphs.
A split graph is a graph that admits a partition C,I of its set of vertices such that C is a clique and I an independent set of vertices. Thus, a split graph is a particular case of a starlike graph, in which each D_{
i
} is a singleton, for 1≤i≤s.
This is also joint work with Cerioli [17]. In this paper, the authors show that the class of starlike (starlikethreshold) graphs contains the class of edge clique graphs of generalized starlike (starlikethreshold) graphs. In addition, every starlike (starlikethreshold) graph which is an edge clique graph is an edge clique graph of a generalized starlike (starlikethreshold) graph. They also prove that a starlikethreshold graph is an edge clique graph if and only if its maximal cliques and intersections of maximal cliques are triangular sets.
Directed graphs
Jayme published several papers related to efficient algorithms for directed graphs. Let us take a brief look at each one of them.
Enumeration of directed circuits
This is a joint work with Lauer [54]. The authors give an O(n+mc) algorithm for enumerating all the directed circuits of a directed graph on m edges, n vertices, and c directed circuits.
Enumeration of kernels
A kernelN of a directed graph D is an independent set of vertices of D such that for every w∈V(D)−N there is an edge from w to N. The existence of a kernel in an directed graph with no odd directed cycles was proved by Richardson [41].
This is a joint work with Chaty [53]. The authors give an algorithm for generating all distinct kernels in a directed graph D with no odd directed circuits. The complexity of the algorithm is O(nm(k+1)), where n, m, and k are the number of vertices, edges, and kernels of D. Also, they show that the problem of determining the number of kernels in a directed graph D is #Pcomplete, even if the longest directed circuit of D has length two.
A minimax equality
The problem of finding the minimum set of vertices that intersects all circuits in a directed graph is NPcomplete [21]. Jayme published a paper [47] in which he introduces the class of connectively reducible digraphs and shows that it contains two classes known to admit polynomial solutions: the class of fully reducible subgraphs and the class of cyclically reducible digraphs. He also describes an algorithm O(n^{2}(n+m)) that recognizes connectively reducible directed graphs and determines a (minimum) set T of vertices that intersects all directed circuits for those graphs and a (maximum) vertexdisjoint set of directed circuits having cardinality equal to that of T.
Orientations with single source and sink
This is joint work with Persiano and Oliveira [57]. Given an undirected graph G, possibly with multiple edges, and distinct vertices s and t of G, the authors consider several orientations D of G. One of these orientations is acyclic and has s and t as the only source and sink of D, respectively. They show that this is possible if and only if graph G+st is 2connected. For each of the problems considered, they use depthfirst search to give linear time algorithms for finding the orientations or determine that they do not exist.
Generation of acyclic orientations
This is joint work with Barbosa [4]. The authors describe an algorithm for finding all the acyclic orientations of a graph G in overall time O((n+m)α) and delay complexity O(n(n+m)), where G has n vertices, m edges, and α acyclic orientations. The space required is O(n+m).
Rooted tree structure
A directed graph D=D(V,E) with a given root vertex s is reducible if every depthfirst search tree with root s has the same set B of back edges. Thus, for a reducible directed graph D, the associated dag (the subgraph with vertex set V and edge set E−B) is uniquely defined. A tree reducible graph is a reducible subgraph for which the transitive reduction (a smallest directed graph with the same reachability) of the associated dag is an arborescence (outdirected tree) with root s.
In this paper [44], Jayme gives the polynomial algorithm for (1) recognizing, (2) finding isomorphisms between, and (3) finding minimum equivalent directed graphs for tree reducible graphs.
Splitindifference graphs
This is a joint work with Ortiz and Maculan [33]. An indifference graph is an intersection graph on a set of unit intervals on the real line. A splitindifference graph is a split graph that is also an indifference graph. The authors give the following characterization of splitindifference graphs.
Theorem 9
A connected graphGis splitindifference if and only if

(i)
Gis complete, or

(ii)
Gis the union of two cliquesG_{1}andG_{2}such thatG_{1}−G_{2}=K_{1}, or

(iii)
Gis the union of three cliquesG_{1}, G_{2}, G_{3}such thatG_{1}−G_{2}=K_{1}, G_{2}−G_{3}=K_{1}and
$$V(G_1) \cap V(G_3) = \emptyset\quad\mbox{or} \quad V(G_1) \cup V(G_3) = V(G).$$
Using that characterization, they determine the chromatic index χ′(G) of splitindifference graphs. In order to do that, they construct an edge coloring of K_{2n}, n≥3, using 2n−1 colors such that K_{2n} has a perfect matching without color repetitions.
The resulting algorithm is very simple. It determines in linear time an optimum edge coloring of a splitindifference graph.
Other results
There are many other results that I should describe, but length restrictions force me to be very concise.
Task scheduling
Jayme has four papers in this area, three of them with Błażewicz and Kubiak [6–8, 45].
Euler tours
A joint work with Cáceres, Deo, and Sastry [13] describes an alternative implementation of Atallah and Vishkin’s parallel algorithm for finding an Euler tour of a graph [1].
Search
I should mention here three papers. The first paper is a joint work with Wilson, on ternary trees [56]. The second paper is joint work with Navarro et al., on optimal binary search trees with costs depending on the access paths [59].
The third paper is a joint work with Moscarini and Petreschi, Node Searching and Starlike Graphs. It is a very interesting paper. Let G be a graph whose vertices are contaminated. Assigning a searcher to a contaminated vertex makes it become guarded. Removing the searcher of a guarded vertex turns it clear. However, a clear vertex becomes contaminated again if it has a contaminated neighbor. The nodesearch number of G is the least number of searchers needed to clear all its vertices. Gustedt [23] has shown that the problem of determining the node search number of G is NPhard for uniform kstarlike graphs. These graphs are generalizations of split graphs, obtained when each vertex of the independent set of the bipartition of the split graph is replaced by a kvertex clique. The authors describe necessary and sufficient conditions for finding the nodesearch number of a uniform kstarlike graph. The characterization described extends a corresponding result for split graphs by Kloks [27]. In addition, it leads to a new algorithm for finding the nodesearch number for graphs of this class.