Realizability of Graphs

A graph is d-realizable if, for every configuration of its vertices in E^N, there exists a another corresponding configuration in E^d with the same edge lengths. A graph is 2-realizable if and only if it is a partial 2-tree, i.e., a subgraph of the 2-sum of triangles in the sense of graph theory. We show that a graph is 3-realizable if and only if it does not have K₅ or the 1-skeleton of the octahedron as a minor.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

On simplices in diameter graphs in ℝ4

Agraph G is a diameter graph in ?^d if its vertex set is a finite subset in ?^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

     

On the dimension to represent a graph by a unit distance graph

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR ^2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a unit distance graph inR ^2d.     

Regular pseudo-median graphs

A graph is pseudo-median if for every triple u, v, w of vertices there exists either a unique vertex between each pair of them (if their mutual distances sum up to an even number) or a unique triangle whose edges lie between the three pairs of u, v, w, respectively (if the distance sum is odd). We show that a finite pseudo-median graph is regular if and only if it is the Cartesian product of a hypercube with either a complete graph or a hyper-octahedron. Every self-map of a pseudo-median graph that preserves or collapses edges has an invariant regular pseudo-median subgraph. Furthermore, the set of all vertices minimizing the total distance to the vertices of a pseudo-median graph induces a regular pseudo-median subgraph.     

A characterization of graphs without long induced paths

In a connected graph define the k-center as the set of vertices whose distance from any other vertex is at most k. We say that a vertex set S d-dominates G if for every vertex x there is a y ∈ S whose distance from x is at most d. Call a graph P_t-free if it does not contain a path on t vertices as an induced subgraph. We prove that a connected graph is P_2k-1-free (P_2k-free) if and only if each of its connected induced subgraphs H satisfy the following property: The k-center of H (k - 1)-dominates ((k - 2)-dominates) H. Moreover, we show that the subgraph induced by the (t - 3)-center in any P_t-free connected graph is again connected and has diameter at most t - 3.     

Computing solutions for matching games

A matching game is a cooperative game (N, v) defined on a graph G = (N, E) with an edge weighting w: E? \mathbb R₊{w: E\to {\mathbb R}_+}. The player set is N and the value of a coalition S í N{S \subseteq N} is defined as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm + n ² log n) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core member if the core is nonempty. This algorithm improves previous work based on the ellipsoid method and can also be used to compute stable solutions for instances of the stable roommates problem with payments. Second we show that the nucleolus of an n-player matching game with a nonempty core can be computed in O(n ⁴) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we prove that is NP-hard to determine an imputation with minimum number of blocking pairs, even for matching games with unit edge weights, whereas the problem of determining an imputation with minimum total blocking value is shown to be polynomial-time solvable for general matching games.     

Embedding arbitrary finite simple graphs into small strongly regular graphs

It is well known that any finite simple graph Γ is an induced subgraph of some exponentially larger strongly regular graph Γ (e.g., [2, 8]). No general polynomial‐size construction has been known. For a given finite simple graph Γ on υ vertices, we present a construction of a strongly regular graph Γ on O(υ⁴) vertices that contains Γ as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 1–8, 2000     

Characterizing subgraphs of Hamming graphs

Cartesian products of complete graphs are known as Hamming graphs. Using embeddings into Cartesian products of quotient graphs we characterize subgraphs, induced subgraphs, and isometric subgraphs of Hamming graphs. For instance, a graph G is an induced subgraph of a Hamming graph if and only if there exists a labeling of E(G) fulfilling the following two conditions: (i) edges of a triangle receive the same label; (ii) for any vertices u and v at distance at least two, there exist two labels which both appear on any induced u, υ‐path. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 302–312, 2005     

Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.     

Large cliques or stable sets in graphs with no four‐edge path and no five‐edge path in the complement

Erd?s and Hajnal [Discrete Math 25 (1989), 37–52] conjectured that, for any graph H, every graph on n vertices that does not have H as an induced subgraph contains a clique or a stable set of size n^?(H) for some ?(H)>0. The Conjecture 1. known to be true for graphs H with |V(H)|≤4. One of the two remaining open cases on five vertices is the case where H is a four‐edge path, the other case being a cycle of length five. In this article we prove that every graph on n vertices that does not contain a four‐edge path or the complement of a five‐edge path as an induced subgraph contains either a clique or a stable set of size at least n^1/6. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Minimally Non-Preperfect Graphs of Small Maximum Degree

 A graph G is called preperfect if each induced subgraph G ^′⊆G of order at least 2 has two vertices x, y such that either all maximum cliques of G ^′ containing x contain y, or all maximum independent sets of G ^′ containing y contain x, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199–208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: (i) A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. (ii) If a graph G is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if G is bipartite, 3-edge-connected, regular of degree d for some d≥3, and contains no 3-edge-connected d ^′-regular subgraph for any 3≤d ^′<d. Received: March 4, 1998 Final version received: August 14, 1999     

Optimizing weakly triangulated graphs

A graph is weakly triangulated if neither the graph nor its complement contains a chordless cycle with five or more vertices as an induced subgraph. We use a new characterization of weakly triangulated graphs to solve certain optimization problems for these graphs. Specifically, an algorithm which runs inO((n + e)n ³) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems. Also, anO((n + e)n ⁴) time algorithm is presented which solves the weighted versions of these problems.The author acknowledges the support of an N.S.E.R.C. Canada postgraduate scholarship.The author acknowledges the support of the U.S. Air Force Office of Scientific Research under grant number AFOSR 0271 to Rutgers University.     

A sparse graph almost as good as the complete graph on points inK dimensions

A setV ofn points ink-dimensional space induces a complete weighted undirected graph as follows. The points are the vertices of this graph and the weight of an edge between any two points is the distance between the points under someL _p metric. Let ε≤1 be an error parameter and letk be fixed. We show how to extract inO(n logn+ε ^−k log(1/ε)n) time a sparse subgraphG=(V, E) of the complete graph onV such that: (a) for any two pointsx, y inV, the length of the shortest path inG betweenx andy is at most (1+∈) times the distance betweenx andy, and (b)|E|=O(ε^−k n).     

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog^1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog^1/6 n vertices.     

Contact patterns of equal nonoverlapping spheres

LetF be a set of nonoverlapping spheres in Euclideann-spaceE ⁿ . By the contact pattern ofF we mean the graph whose vertex set isF and two vertices are adjacent whenever the corresponding spheres touch each other. Every graph turns out to be a contact pattern in some dimension. This paper studies the smallest dimensionn for a graphG such thatG is a contact pattern inE ⁿ . Among others, the smallest dimensions are determined for the join of a large complete graph and an empty graph, and for complete multipartite graphs with more vertex classes than the size of its largest vertex class.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

On graphs with prescribed median I

The distance of a vertex u in a connected graph H is the sum of all the distances from u to the other vertices of H. The median M(H) of H is the subgraph of H induced by the vertices of minimum distance. For any graph G, let f(G) denote the minimum order of a connected graph H satisfying M(H) ? G. It is shown that if G has n vertices and minimum degree δ then f(G) ? 2n ? δ + 1. Graphs having both median and center prescribed are constructed. It is also shown that if the vertices of a K_r are removed from a graph H, then at most r components of the resulting graph contain median vertices of H.     

Extremal graphs of diameter at most 6 after deleting any vertex

Suppose G is a graph of n vertices and diameter at most d having the property that, after deleting any vertex, the resulting subgraph has diameter at most 6. Then G contains at least max{n, (4n - 8)/3} edges if 4 ≤ d ≤ 6.     

What is the furthest graph from a hereditary property?

For a graph property P, the edit distance of a graph G from P, denoted E_P(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of E_P(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = E_P(G(n,p(P))) + o(n²) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008