Reconstruction of interval graphs

The graph reconstruction conjecture is a long-standing open problem in graph theory. There are many algorithmic studies related to it, such as DECK CHECKING, LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING. We study these algorithmic problems by limiting the graph class to interval graphs. Since we can solve GRAPH ISOMORPHISM for interval graphs in polynomial time, DECK CHECKING for interval graphs is easily done in polynomial time. Since the number of interval graphs that can be obtained from an interval graph by adding a vertex and edges incident to it can be exponentially large, developing polynomial time algorithms for LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING on interval graphs is not trivial. We present that these three problems are solvable in polynomial time on interval graphs.     

On the number of Eulerian orientations of a graph

An Eulerian orientation of an undirected Eulerian graph is an orientation of the edges of the graph such that for every vertex the in-degree is equal to the out-degree. Eulerian orientations are natural flow-like structures, and Welsh has pointed out that computing their number corresponds to evaluating the Tutte polynomial at the point (0, –2) [JVW], [Wl], and is further equivalent to evaluating ice-type partition functions in statistical physics [W2]. In this paper we resolve the complexity of counting the number of Eulerian orientations of an arbitrary Eulerian graph.We give an efficient randomized approximation algorithm for counting Eulerian orientations of any Eulerian graph. Our algorithm is based on a reduction to counting perfect matchings for a class of graphs for which the methods of Broder [B], Jerrum and Sinclair [JS1], and others [DL] [DS] apply. A crucial step of the reduction is the Monotonicity Lemma (Lemma 3.1) which is of independent combinatorial interest. Roughly speaking, the Monotonicity Lemma establishes the intuitive fact that increasing the number of constraints applied on a flow problem cannot increase the number of solutions. The proof of the lemma involves a new decomposition technique which decouples problematically overlapping structures (a recurrent obstacle in handling large combinatorial populations) and allows detailed enumeration arguments. As a by-product, we exhibit a class of graphs for which perfect and near-perfect matchings are polynomially related, and hence the permanent can be approximated, for reasons other than short augmenting paths (previously the only known approach).We also give the complementary hardness result, namely, that counting exactly Eulerian orientations is #P-complete. Finally, we provide some connections with counting Euler tours.     

Packing triangles in bounded degree graphs

We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectively 5) or more.     

De Bruijn sequences and De Bruijn graphs for a general language

A de Bruijn sequence over a finite alphabet of span n is a cyclic string such that all words of length n appear exactly once as factors of this sequence. We extend this definition to a subset of words of length n, characterizing for which subsets exists a de Bruijn sequence. We also study some symbolic dynamical properties of these subsets extending the definition to a language defined by forbidden factors. For these kinds of languages we present an algorithm to produce a de Bruijn sequence. In this work we use graph-theoretic and combinatorial concepts to prove these results.     

Symbolic graphs for attributed graph constraints

In this paper we present a new class of graphs, called symbolic graphs, to define a new class of constraints on attributed graphs. In particular, in the first part of the paper, we study the category of symbolic graphs showing that it satisfies some properties, which are the basis for the work that we present in the second part of the paper, where we study how to reason with attributed graph constraints. More precisely, we define a set of inference rules, which are the instantiation of the inference rules defined in a previous paper, for reasoning about constraints on standard graphs, showing their soundness and (weak) completeness. Moreover, the proof of soundness and completeness is also an instantiation of the corresponding proof for standard graph constraints, using the categorical properties studied in the first part of the paper. Finally, we show that adding a new inference rule makes our system sound and strongly complete.     

Drawing slicing graphs with face areas

We consider orthogonal drawings of a plane graph G$G$ with specified face areas. For a natural number k$k$, a k$k$-gonal drawing of G$G$ is an orthogonal drawing such that the boundary of G$G$ is drawn as a rectangle and each inner face is drawn as a polygon with at most k$k$ corners whose area is equal to the specified value. In this paper, we show that every slicing graph G$G$ with a slicing tree T$T$ and a set of specified face areas admits a 10-gonal drawing D$D$ such that the boundary of each slicing subgraph that appears in T$T$ is also drawn as a polygon with at most 10 corners. Such a drawing D$D$ can be found in linear time.     

Degree-constrained decompositions of graphs: Bounded treewidth and planarity

We study the problem of decomposing the vertex set V$V$ of a graph into two nonempty parts _V1,_V2$V_1,V_2$ which induce subgraphs where each vertex v∈_V1$v\in V_1$ has degree at least a(v)$a(v)$ inside _V1$V_1$ and each v∈_V2$v\in V_2$ has degree at least b(v)$b(v)$ inside _V2$V_2$. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This result and its variants are then applied to designing polynomial-time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible.     

On acyclic edge coloring of toroidal graphs

Let c be a proper edge coloring of a graph G. If there exists no bicolored cycle in G with respect to c, then c is called an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ and girth g. In Dong and Xu (2010) [8], Dong and Xu proved that G admits an acyclic edge coloring with Δ(G) colors if Δ?8 and g?7, or Δ?6 and g?8, or Δ?5 and g?9, or Δ?4 and g?10, or Δ?3 and g?14. In this note, we fix a small gap in the proof of Dong and Xu (2010) [8], and generalize the above results to toroidal graphs.     

Node overlap removal in clustered directed acyclic graphs

Graph drawing and visualization represent structural information as diagrams of abstract graphs and networks. An important subset of graphs is directed acyclic graphs (DAGs). This paper presents a new E-Spring algorithm, extended from the popular spring embedder model, which eliminates node overlaps in clustered DAGs. In this framework, nodes are modeled as non-uniform charged particles with weights, and a final drawing is derived by adjusting the positions of the nodes according to a combination of spring forces and repulsive forces derived from electrostatic forces between the nodes. The drawing process needs to reach a stable state when the average distances of separation between nodes are near optimal. We introduce a stopping condition for such a stable state, which reduces equilibrium distances between nodes and therefore results in a significantly reduced area for DAG visualization. It imposes an upper bound on the repulsive forces between nodes based on graph geometry. The algorithm employs node interleaving to eliminate any residual node overlaps. These new techniques have been validated by visualizing eBay buyer–seller relationships and has resulted in overall area reductions in the range of 45–79%.     

Coloring inductive graphs on-line

In this paper we consider the problem of on-line graph coloring. In an instance of on-line graph coloring, the nodes are presented one at a time. As each node is presented, its edges to previously presented nodes are also given. Each node must be assigned a color, different from the colors of its neighbors, before the next node is given. LetA(G) be the number of colors used by algorithmA on a graphG and letx(G) be the chromatic number ofG. The performance ratio of an on-line graph coloring algorithm for a class of graphsC is max_{G C(A(G)/(G))}. We consider the class ofd-inductive graphs. A graphG isd-inductive if the nodes ofG can be numbered so that each node has at mostd edges to higher-numbered nodes. In particular, planar graphs are 5-inductive, and chordal graphs arex(G)-inductive. First Fit is the algorithm that assigns each node the lowest-numbered color possible. We show that ifG isd-inductive, then First Fit usesO(d logn) colors onG. This yields an upper bound ofo(logn) on the performance ratio of First Fit on chordal and planar graphs. First Fit does as well as any on-line algorithm ford-inductive graphs: we show that, for anyd and any on-line graph coloring algorithmA, there is ad-inductive graph that forcesA to use (d logn) colors to colorG. We also examine on-line graph coloring with lookahead. An algorithm is on-line with lookaheadl, if it must color nodei after examining only the firstl+i nodes. We show that, forl/logn, the lower bound ofd logn colors still holds.This research was supported by an IBM Graduate Fellowship.     

Simplicity in Eulerian circuits: Uniqueness and safety

An Eulerian circuit in a directed graph is one of the most fundamental Graph Theory notions. Detecting if a graph G has a unique Eulerian circuit can be done in polynomial time via the BEST theorem by de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte (1941–1951) [15], [16] (involving counting arborescences), or via a tailored characterization by Pevzner, 1989 (involving computing the intersection graph of simple cycles of G), both of which thus rely on overly complex notions for the simpler uniqueness problem.In this paper we give a new linear-time checkable characterization of directed graphs with a unique Eulerian circuit. This is based on a simple condition of when two edges must appear consecutively in all Eulerian circuits, in terms of cut nodes of the underlying undirected graph of G. As a by-product, we can also compute in linear-time all maximal safe walks appearing in all Eulerian circuits, for which Nagarajan and Pop proposed in 2009 [12] a polynomial-time algorithm based on Pevzner characterization.     

New spectral lower bounds on the bisection width of graphs

The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled using the graph bisection problem. The spectral lower bound of λ₂|V|/4 for the bisection width of a graph is widely known. The bisection width is equal to λ₂|V|/4 iff all vertices are incident to λ₂/2 cut edges in every optimal bisection.

We present a new method of obtaining tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. We define some global expansion properties and we show that the spectral lower bound increases with this global expansion. Under certain conditions we obtain a lower bound depending on λ₂^β|V| with . We also present examples of graphs for which our new bounds are tight up to a constant factor. As a by-product, we derive new lower bounds for the bisection widths of 3- and 4-regular Ramanujan graphs.     

DNA physical mapping and alternating Eulerian cycles in colored graphs

Small-scale DNA physical mapping (such as the Double Digest Problem or DDP) is an important and difficult problem in computational molecular biology. When enzyme sites are modeled by a random process, the number of solutions to DDP is known to increase exponentially as the length of DNA increases. However, the overwhelming majority of solutions are very similar and can be transformed into each other by simple transformations. Recently, Schmitt and Waterman [SW] introduced equivalence classes on the set of DDP solutions and raised an open problem to completely characterize equivalent physical maps.We study the combinatorics of multiple solutions and the cassette transformations of Schmitt and Waterman. We demonstrate that the solutions to DDP are closely associated with alternating Eulerian cycles in colored graphs and study order transformations of alternating cycles. We prove that every two alternating Eulerian cycles in a bicolored graph can be transformed into each other by means of order transformations. Using this result we obtain a complete characterization of equivalent physical maps in the Schmitt-Waterman problem. It also allows us to prove Ukkonen's conjecture on word transformations preservingq-gram composition.This research was supported in part by the National Science Foundation under Grants DMS 90-05833 and CCR-93-08567 and the National Institute of Health under Grant GM-36230.     

Some optimization problems on weak-bisplit graphs

A new decomposition scheme for bipartite graphs namely canonical decomposition was introduced by Fouquet et al. [Internat. J. Found. Comput. Sci. 10 (1999) 513-533]. The so-called weak-bisplit graphs are totally decomposable following this decomposition. We present here some optimization problems for general bipartite graphs which have efficient solutions when dealing with weak-bisplit graphs.     

The diameter of Hanoi graphs

Many questions regarding the Tower of Hanoi problem have been posed and answered during the years. Variants of the classical puzzle, such as allowing more than 3 pegs, and imposing limitations on the possible moves among the pegs, raised the analogous questions for those variants. One such question is: given a variant, and a certain number of disks, find a pair of disk arrangements such that the minimal number of moves required for changing from the first to the second is maximal over all pairs. One of the main results of the paper is identifying these for the Cyclic_h variants—the variants with h pegs arranged along a uni-directional circle—to be the pairs of perfect configurations where the destination peg is right before the source peg.     

Signed and minus clique-transversal functions on graphs

A minus (respectively, signed) clique-transversal function of a graph G=(V,E) is a function (respectively, {−1,1}) such that _∑u∈Cf(u)?1 for every maximal clique C of G. The weight of a minus (respectively, signed) clique-transversal function of G is f(V)=_∑v∈Vf(v). The minus (respectively, signed) clique-transversal problem is to find a minus (respectively, signed) clique-transversal function of G of minimum weight. In this paper, we present a unified approach to these two problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. We also prove that the signed clique-transversal problem is NP-complete for chordal graphs and planar graphs.     

Simplicial powers of graphs

In a graph, a vertex is simplicial if its neighborhood is a clique. For an integer k≥1, a graph G=(V_G,E_G) is the k-simplicial power of a graph H=(V_H,E_H) (H a root graph of G) if V_G is the set of all simplicial vertices of H, and for all distinct vertices x and y in V_G, xyE_G if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k≤5, k-leaf powers can be recognized in linear time, and for k≤4, structural characterizations are known. For k≥6, the recognition and characterization problems of k-leaf powers are still open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs.     

The clique-transversal set problem in claw-free graphs with degree at most 4

A clique of a graph G is defined as a complete subgraph maximal under inclusion and having at least two vertices. A clique-transversal set D of G is a subset of vertices of G such that D meets all cliques of G. The clique-transversal set problem is to find a minimum clique-transversal set of G. In this paper we present a polynomial time algorithm for the clique-transversal set problem on claw-free graphs with degree at most 4.