On α-redundant vertices in P5-free graphs

We prove that Maximum Stable Set can be solved in polynomial time on two new subclasses of P₅-free graphs, extending some known polynomially solvable cases.     

Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K_1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of Dominating Set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general.The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K_i,j-free graphs, which exclude, for some fixed i,j∈N, the complete bipartite graph K_i,j as a subgraph. For i,j≥2, the class of claw-free graphs and any class of K_i,j-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable.We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t≥4, the Dominating Set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K_1,t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W[2]-hard in these graph classes.Finally, we show that for any t∈N, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.     

Fault-tolerant hamiltonian laceability of hypercubes

It is known that every hypercube Q_n is a bipartite graph. Assume that n?2 and F is a subset of edges with |F|?n−2. We prove that there exists a hamiltonian path in Q_n−F between any two vertices of different partite sets. Moreover, there exists a path of length 2ⁿ−2 between any two vertices of the same partite set. Assume that n?3 and F is a subset of edges with |F|?n−3. We prove that there exists a hamiltonian path in Q_n−{v}−F between any two vertices in the partite set without v. Furthermore, all bounds are tight.     

Counting the Number of Perfect Matchings in K5-Free Graphs

Counting the number of perfect matchings in graphs is a computationally hard problem. However, in the case of planar graphs, and even for K_3,3-free graphs, the number of perfect matchings can be computed efficiently. The technique to achieve this is to compute a Pfaffian orientation of a graph. In the case of K₅-free graphs, this technique will not work because some K₅-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K₅-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC². We remark that our results generalize to graphs without singly-crossing minor.     

Property Testing in Bounded Degree Graphs

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far'' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/? , always accept the graph when it has the tested property, and reject with high probability if the graph is ? -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ? dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Pancyclicity and bipancyclicity of conditional faulty folded hypercubes

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network’s topology can simulate rings of various lengths. An n-dimensional folded hypercube FQ_n is an attractive variant of an n-dimensional hypercube Q_n that is obtained by establishing some extra edges between the vertices of Q_n. FQ_n for any odd n is known to be bipartite. In this paper, we explore the pancyclicity and bipancyclicity of FQ_n. For any FQ_n (n ? 2) with at most 2n − 3 faulty edges, where each vertex is incident to at least two fault-free edges, we prove that there exists a fault-free cycle of every even length from 4 to 2ⁿ; and when n ? 2 is even, we prove there also exists a fault-free cycle of every odd length from n + 1 to 2ⁿ − 1. The result is optimal with respect to the number of faulty edges tolerated.     

Colored Simultaneous Geometric Embeddings and?Universal Pointsets

Universal pointsets can be used for visualizing multiple relationships on the same set of objects or for visualizing dynamic graph processes. In simultaneous geometric embeddings, the same point in the plane is used to represent the same object as a way to preserve the viewer??s mental map. In colored simultaneous embeddings this restriction is relaxed, by allowing a given object to map to a subset of points in the plane. Specifically, consider a set of graphs on the same set of n vertices partitioned into k colors. Finding a corresponding set of k-colored points in the plane such that each vertex is mapped to a point of the same color so as to allow a straight-line plane drawing of each graph is the problem of colored simultaneous geometric embedding. For n-vertex paths, we show that there exist universal pointsets of size n, colored with two or three colors. We use this result to construct colored simultaneous geometric embeddings for a 2-colored tree together with any number of 2-colored paths, and more generally, a?2-colored outerplanar graph together with any number of 2-colored paths. For n-vertex trees, we construct small near-universal pointsets for 3-colored caterpillars of size n, 3-colored radius-2 stars of size n+3, and 2-colored spiders of size n. For n-vertex outerplanar graphs, we show that these same universal pointsets also suffice for 3-colored K ₃-caterpillars, 3-colored K ₃-stars, and 2-colored fans, respectively. We also present several negative results, showing that there exist a 2-colored planar graph and pseudo-forest, three 3-colored outerplanar graphs, four 4-colored pseudo-forests, three 5-colored pseudo-forests, five 5-colored paths, two 6-colored biconnected outerplanar graphs, three 6-colored cycles, four 6-colored paths, and three 9-colored paths that cannot be simultaneously embedded.     

A constant approximation algorithm for the densest k-subgraph problem on chordal graphs

The densest k-subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The DkS problem is NP-hard even for special graph classes including bipartite, planar, comparability and chordal graphs, while no constant approximation algorithm is known for any of these classes. In this paper we present a 3-approximation algorithm for the class of chordal graphs. The analysis of our algorithm is based on a graph theoretic lemma of independent interest.     

Online Coloring of Bipartite Graphs with and without Advice

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ?1.13746?log₂(n)?0.49887? colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ?log₂ n?+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that \(O(\sqrt{n})\) advice bits are sufficient for coloring any bipartite graph on n vertices with at most ?log₂ n? colors.     

Approximating maximum weight K-colorable subgraphs in chordal graphs

We present a 2-approximation algorithm for the problem of finding the maximum weight K-colorable subgraph in a given chordal graph with node weights. The running time of the algorithm is O(K(n+m)), where n and m are the number of vertices and edges in the given graph.     

Finding Maximum Induced Matchings in Subclasses of Claw-Free and <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs,and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P ₅,D _m )-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.     

Hamiltonian laceability of bubble-sort graphs with edge faults

It is known that the n-dimensional bubble-sort graph B_n is bipartite, (n − 1)-regular, and has n! vertices. We first show that, for any vertex v, B_n − v has a hamiltonian path between any two vertices in the same partite set without v. Let F be a subset of edges of B_n. We next show that B_n − F has a hamiltonian path between any two vertices of different partite sets if ∣F∣ is at most n − 3. Then we also prove that B_n − F has a path of length n! − 2 between any pair of vertices in the same partite set.     

A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs

We study a problem of lower bounds on straight line drawings of planar graphs. We show that at least 1.235·n points in the plane are required to draw each n-vertex planar graph with edges drawn as straight line segments (for sufficiently large n). This improves the previous best bound of 1.206·n (for sufficiently large n) due to Chrobak and Karloff [Sigact News 20 (4) (1989) 83-86]. Our contribution is twofold: we improve the lower bound itself and we give a significantly simpler and more straightforward proof.     

Updating the complexity status of coloring graphs without a fixed induced linear forest

A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph P_k denotes a path on k vertices. The ?-Coloring problem is the problem to decide whether a graph can be colored with at most ? colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P₈-free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-Coloring is NP-complete for P₉-free graphs, and a result of Woeginger and Sgall, who showed that 5-Coloring is NP-complete for P₈-free graphs. Additionally, we prove that the precoloring extension version of 4-Coloring is NP-complete for P₇-free graphs, but that the precoloring extension version of 3-Coloring can be solved in polynomial time for (P₂+P₄)-free graphs, a subclass of P₇-free graphs. Here P₂+P₄ denotes the disjoint union of a P₂ and a P₄. We denote the disjoint union of s copies of a P₃ by sP₃ and involve Ramsey numbers to prove that the precoloring extension version of 3-Coloring can be solved in polynomial time for sP₃-free graphs for any fixed s. Combining our last two results with known results yields a complete complexity classification of (precoloring extension of) 3-Coloring for H-free graphs when H is a fixed graph on at most 6 vertices: the problem is polynomial-time solvable if H is a linear forest; otherwise it is NP-complete.     

Antimagic labeling and canonical decomposition of graphs

An antimagic labeling of a connected graph with m edges is an injective assignment of labels from {1,…,m} to the edges such that the sums of incident labels are distinct at distinct vertices. Hartsfield and Ringel conjectured that every connected graph other than K₂ has an antimagic labeling. We prove this for the classes of split graphs and graphs decomposable under the canonical decomposition introduced by Tyshkevich. As a consequence, we provide a sufficient condition on graph degree sequences to guarantee an antimagic labeling.     

Almost Exact Matchings

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)?1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K _3,3-minor free graphs (these include all planar graphs as well as many others) in O(n ^3.19) worst case time. Our algorithm can also count the number of perfect matchings in K _3,3-minor free graphs in O(n ^2.19) time.     

Structural Properties of Context-Free Sets of Graphs Generated by Vertex Replacement

We establish that a Vertex Replacement set of graphs, i.e., a set of graphs generated by a C-edNCE or, equivalently, by a separated handle rewriting graph grammar is Hyperedge Replacement, i.e., is generated by a hyperedge replacement graph grammar, iff its graphs do not contain arbitrary large complete bipartite graphs K_{n, n} as subgraphs. Another equivalent condition is that its graphs have a number of edges that is linearly bounded in terms of the number of vertices. These properties are decidable by means of an appropriate extension of the theorem by Parikh that characterizes the commutative images of context-free languages. We extend these results to hypergraphs.     

Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs

The densest k-subgraph problem asks for a k-vertex subgraph with the maximum number of edges. This problem is NP-hard on bipartite graphs, chordal graphs, and planar graphs. A 3-approximation algorithm is known for chordal graphs. We present -approximation algorithms for proper interval graphs and bipartite permutation graphs. The latter result relies on a new characterisation of bipartite permutation graphs which may be of independent interest.     

On simultaneous straight-line grid embedding of a planar graph and its dual

Simultaneous representations of planar graphs and their duals normally require that the dual vertices to be placed inside their corresponding primal faces, and the edges of the dual graph to cross only their corresponding primal edges. Erten and Kobourov [C. Erten, S.G. Kobourov, Simultaneous embedding of a planar graph and its dual on the grid, Theory Computer Systems 38 (2005) 313-327] provided a linear time algorithm on simultaneous straight-line grid embedding of a 3-connected planar graph and its dual such that all the vertices are placed on grid points and each edge is drawn as one straight-line segment except for one which is drawn using two segments. Their drawing size is (2n−2)×(2n−2), where n is the total number of vertices in the graph and its dual. They raised an open question on whether there is a large class of planar graphs that allows this simultaneous straight-line grid embedding on a smaller grid. We answer this open question by giving a linear time simultaneous straight-line grid embedding algorithm for a 3-connected planar graph and its dual on a grid of size (n−1)×n.     

Bimagic Vertex Labelings

The notion of the equivalence of vertex labelings on a given graph is introduced. The equivalence of three bimagic labelings for regular graphs is proved. A particular solution is obtained for the problem of the existence of a 1-vertex bimagic vertex labeling of multipartite graphs, namely, for graphs isomorphic with K_{n, n, m}. It is proved that the sequence of bi-regular graphs K_n(ij)?=?((K_n???1???M)?+?K₁)???(u_nu_i)???(u_nu_j) admits 1-vertex bimagic vertex labeling, where u_i, u_j is any pair of non-adjacent vertices in the graph K_n???1???M, u_n is a vertex of K₁, M is perfect matching of the complete graph K_n???1. It is established that if an r-regular graph G of order n is distance magic, then graph G + G has a 1-vertex bimagic vertex labeling with magic constants (n?+?1)(n?+?r)/2?+?n² and (n?+?1)(n?+?r)/2?+?nr. Two new types of graphs that do not admit 1-vertex bimagic vertex labelings are defined.