On second order degree of graphs

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v.In this paper we address the following question:What are the sufficient conditions for a graph to have a vertex v such that d2(v) ≥ d(v),where d(v) denotes the degree of v? Among other results,every graph of minimum degree exactly 2,except four graphs,is shown to have a vertex of second order degree as large as its own degree.Moreover,every K-4-free graph or every maximal planar graph is shown to have a vertex v such that d2(v) ≥ d(v).Other sufficient conditions on graphs for guaranteeing this property are also proved.     

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

d , and the testing algorithm can perform queries of the form: ``who is the ith neighbor of vertex v'. The tester should determine with high probability whether the graph is bipartite or ε-far from bipartite for any given distance parameter ε. Distance between graphs is defined to be the fraction of entries on which the graphs differ in their incidence-lists representation. Our testing algorithm has query complexity and running time where N is the number of graph vertices. It was shown before that queries are necessary (for constant ε), and hence the performance of our algorithm is tight (in its dependence on N), up to polylogarithmic factors. In our analysis we use techniques that were previously applied to prove fast convergence of random walks on expander graphs. Here we use the contrapositive statement by which slow convergence implies small cuts in the graph, and further show that these cuts have certain additional properties. This implication is applied in showing that for any graph, the graph vertices can be divided into disjoint subsets such that: (1) the total number of edges between the different subsets is small; and (2) each subset itself exhibits a certain mixing property that is useful in our analysis. Received: February 6, 1998     

Distance and routing labeling schemes for non-positively curved plane graphs

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently (e.g., in constant or logarithmic time) by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes are schemes that label the vertices of a graph with short labels in such a way that given the label of a source vertex and the label of a destination, it is possible to compute efficiently (e.g., in constant or logarithmic time) the port number of the edge from the source that heads in the direction of the destination. In this paper we show that the three major classes of non-positively curved plane graphs enjoy such distance and routing labeling schemes using O(log²n) bit labels on n-vertex graphs. In constructing these labeling schemes interesting metric properties of those graphs are employed.     

ON A CONJECTURE ON nTH ORDER DEGREE REGULAR GRAPHS

Abstract

For n a positive integer and v a vertex of a graph G, the nth order degree of v in G, denoted by deg_nv, is the number of vertices at distance n from v. The graph G is said to be nth order regular of degree k if, for every vertex v of G, deg_nv = k. The following conjecture due to Alavi, Lick, and Zou is proved: For n ≥ 2, if G is a connected nth order regular graph of degree 1, then G is either a path of length 2n—1 or G has diameter n. Properties of nth order regular graphs of degree k, k ≥ 1, are investigated.     

Variable neighborhood search for extremal graphs. 22. Extending bounds for independence to upper irredundance

A set of vertices S in a graph G is independent if no neighbor of a vertex of S belongs to S. A set of vertices U in a graph G is irredundant if each vertex v of U has a private neighbor, which may be v itself, i.e., a neighbor of v which is not a neighbor of any other vertex of U. The independence number α (resp. upper irredundance number IR) is the maximum number of vertices of an independent (resp. irredundant) set of G. In previous work, a series of best possible lower and upper bounds on α and some other usual invariants of G were obtained by the system AGX 2, and proved either automatically or by hand. These results are strengthened in the present paper by systematically replacing α by IR. The resulting conjectures were tested by AGX which could find no counter-example to an upper bound nor any case where a lower bound could not be shown to remain tight. Some proofs for the bounds on α carry over. In all other cases, new proofs are provided.     

Local and global average degree in graphs and multigraphs

A vertex v of a graph G is called groupie if the average degree t_v of all neighbors of v in G is not smaller than the average degree t_G of G. Every graph contains a groupie vertex; the problem of whether or not every simple graph on ≧2 vertices has at least two groupie vertices turned out to be surprisingly difficult. We present various sufficient conditions for a simple graph to contain at least two groupie vertices. Further, we investigate the function f(n) = max min_v (t_v/t_G), where the maximum ranges over all simple graphs on n vertices, and prove that f(n) = 1/42n + o(1). The corresponding result for multigraphs is in sharp contrast with the above. We also characterize trees in which the local average degree t_v is constant.     

A Characterization of the degree sequences of 2‐trees

A graph G is a 2‐tree if G = K₃, or G has a vertex v of degree 2, whose neighbors are adjacent, and G/ v is a 2‐ tree. A characterization of the degree sequences of 2‐trees is given. This characterization yields a linear‐time algorithm for recognizing and realizing degree sequences of 2‐trees. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:191‐209, 2008     

Randomized greedy algorithms for finding small k‐dominating sets of regular graphs

A k‐dominating set of a graph G is a subset ?? of the vertices of G such that every vertex of G is either in ?? or at distance at most k from a vertex in ??. It is of interest to find k‐dominating sets of small cardinality. In this paper we consider simple randomized greedy algorithms for finding small k‐dominating sets of regular graphs. We analyze the average‐case performance of the most efficient of these simple heuristics showing that it performs surprisingly well on average. The analysis is performed on random regular graphs using differential equations. This, in turn, proves upper bounds on the size of a minimum k‐dominating set of random regular graphs. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 22, 2005     

Geodeticity of the contour of chordal graphs

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.     

Recoverable values for independent sets

The notion of recoverable value was advocated in the work of Feige, Immorlica, Mirrokni and Nazerzadeh (APPROX 2009) as a measure of quality for approximation algorithms. There, this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has recoverable factor ρ, if for every graph it recovers an independent set of size at least where d(v) is the degree of vertex v, and I ranges over all independent sets in G. Hence, in a sense, from every vertex v in the maximum independent set the algorithm recovers a value of at least toward the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. A simple greedy algorithm achieves . We design a new randomized algorithm for MIS that ensures an expected recoverable factor of at least . In passing, we prove that approximating MIS in graphs with a given k‐coloring within a ratio larger than 2/ k is unique‐games hard. This rules out an alternative approach for obtaining . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 142–159, 2015     

On tree‐decompositions of one‐ended graphs

A graph is one‐ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex v dominates a ray in the end if there are infinitely many paths connecting v to the ray such that any two of these paths have only the vertex v in common. We prove that if a one‐ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree‐decomposition such that the decomposition tree is one‐ended and the tree‐decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one‐ended graph contains an infinite family of pairwise disjoint rays.     

Paired Domination Vertex Critical Graphs

Let γ _pr (G) denote the paired domination number of graph G. A graph G with no isolated vertex is paired domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ _pr (G – v) < γ _pr (G). We call these graphs γ _pr -critical. In this paper, we present a method of constructing γ _pr -critical graphs from smaller ones. Moreover, we show that the diameter of a γ _pr -critical graph is at most and the upper bound is sharp, which answers a question proposed by Henning and Mynhardt [The diameter of paired-domination vertex critical graphs, Czechoslovak Math. J., to appear]. Xinmin Hou: Research supported by NNSF of China (No.10701068 and No.10671191).     

Medians of arbitrary graphs

For each vertex u in a connected graph H, the distance of u is the sum of the distances from u to each of the vertices v of H. A vertex of minimum distance in H is called a median vertex. It is shown that for any graph G there exists a graph H for which the subgraph of H induced by the median vertices is isomorphic to G.     

Vertex fusion under distance constraints

In this paper, we analyze parameter improvement under vertex fusion in a graph G. This is a setting in which a new graph G^′ is obtained after identifying a subset of vertices of G in a single vertex. We are interested in distance parameters, in particular diameter, radius and eccentricity of a vertex v. We show that the corresponding problem is NP-Complete for the three parameters. We also find graph classes in which the problem can be solved in polynomial time.     

Locally s‐distance transitive graphs

We give a unified approach to analyzing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s‐arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s≥2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex‐orbits or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:176‐197, 2012     

Optimal broadcast domination in polynomial time

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that different vertices can be assigned different domination powers. Broadcast domination assigns an integer power f(v)?0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v)?1. The optimal broadcast domination problem seeks to minimize the sum of the powers assigned to the vertices of the graph. Since the presentation of this problem its computational complexity has been open, and the general belief has been that it might be NP-hard. In this paper, we show that optimal broadcast domination is actually in P, and we give a polynomial time algorithm for solving the problem on arbitrary graphs, using a non-standard approach.     

Variable neighborhood search for extremal graphs. 21. Conjectures and results about the independence number

A set of vertices S in a graph G is independent if no neighbor of a vertex of S belongs to S. The independence number α is the maximum cardinality of an independent set of G. A series of best possible lower and upper bounds on α and some other common invariants of G are obtained by the system AGX 2, and proved either automatically or by hand. In the present paper, we report on such lower and upper bounds considering, as second invariant, minimum, average and maximum degree, diameter, radius, average distance, spread of eccentricities, chromatic number and matching number.     

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997     

A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low-weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning treeTusingadoptionsto meet the degree constraints is considered. A novel network-flow-based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraintd(v) for eachvis at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 − min{(d(v) − 2)/(deg_T(v) − 2) : deg_T(v) > 2}, where deg_T(v) is the initial degree ofv. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds foranyalgorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. ChoosingTto be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by variousL_pnorms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.     

A local structure theorem on 5‐connected graphs

An edge of a 5‐connected graph is said to be contractible if the contraction of the edge results in a 5‐connected graph. Let x be a vertex of a 5‐connected graph. We prove that if there are no contractible edges whose distance from x is two or less, then either there are two triangles with x in common each of which has a distinct degree five vertex other than x, or there is a specified structure called a K₄^?‐configuration with center x. As a corollary, we show that if a 5‐connected graph on n vertices has no contractible edges, then it has 2n/5 vertices of degree 5. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 99–129, 2009