On the Structure of Graphs with Bounded Asteroidal Number

 A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d _T (x,y)−d _G (x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G. Received: July 3, 1998 Final version received: August 10, 1999     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

Trees with Equal Domination and Restrained Domination Numbers

Let G = (V,E) be a graph and let S V. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T)=γ_r(T); (ii) T is a γ-excellent tree and T ≠ K₂; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γ_r(T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

A Greedy Partition Lemma for directed domination

A directed dominating set in a directed graph D is a set S of vertices of V such that every vertex u∈V(D)?S has an adjacent vertex v in S with v directed to u. The directed domination number of D, denoted by γ(D), is the minimum cardinality of a directed dominating set in D. The directed domination number of a graph G, denoted Γ_d(G), is the maximum directed domination number γ(D) over all orientations D of G. The directed domination number of a complete graph was first studied by Erd?s [P. Erd?s On a problem in graph theory, Math. Gaz. 47 (1963) 220–222], albeit in a disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if α denotes the independence number of a graph G, we show that α≤Γ_d(G)≤α(1+2ln(n/α)).     

A sufficient condition for bipartite graphs to be type one

The total chromatic number χ_T(G) of graph G is the least number of colors assigned to V(G) ∪ E(G) such that no adjacent or incident elements receive the same color. In this article, we give a sufficient condition for a bipartite graph G to have χ_T(G) = Δ(G) + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 133–137, 1998     

Closed Separator Sets

A smallest separator in a finite, simple, undirected graph G is a set S ⊆ V (G) such that G–S is disconnected and |S|=κ(G), where κ(G) denotes the connectivity of G. A set S of smallest separators in G is defined to be closed if for every pair S,T ∈ S, every component C of G–S, and every component S of G–T intersecting C either X(C,D) := (V (C) ∩ T) ∪ (T ∩ S) ∪ (S ∩ V (D)) is in S or |X(C,D)| > κ(G). This leads, canonically, to a closure system on the (closed) set of all smallest separators of G. A graph H with is defined to be S-augmenting if no member of S is a smallest separator in G ∪ H:=(V (G) ∪ V (H), E(G) ∪ E(H)). It is proved that if S is closed then every minimally S-augmenting graph is a forest, which generalizes a result of Jordán. Several applications are included, among them a generalization of a Theorem of Mader on disjoint fragments in critically k-connected graphs, a Theorem of Su on highly critically k-connected graphs, and an affirmative answer to a conjecture of Su on disjoint fragments in contraction critically k-connected graphs of maximal minimum degree.     

On partitioning the edges of graphs into connected subgraphs

For any positive integer s, an s-partition of a graph G = (V, E) is a partition of E into E₁ ∪ E₂ ∪…? ∪ E_k, where ∣E_i∣ = s for 1 ≤ i ≤ k ? 1 and 1 ≤ ∣E_k∣ ≤ s and each E_i induces a connected subgraph of G. We prove

• (i) If G is connected, then there exists a 2-partition, but not necessarily a 3-partition;
• (ii) If G is 2-edge connected, then there exists a 3-partition, but not necessarily a 4-partition;
• (iii) If G is 3-edge connected, then there exists a 4-partition;
• (iv) If G is 4-edge connected, then there exists an s-partition for all s.

     

Bounds relating the weakly connected domination number to the total domination number and the matching number

Let G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted . A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. In this paper, we show that . Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α^′(G). We also establish that , and show that for every tree T.     

Cochromatic Number and the Genus of a Graph

The cochromatic number of a graph G, denoted by z(G), is the minimum number of subsets into which the vertex set of G can be partitioned so that each sbuset induces an empty or a complete subgraph of G. In this paper we introduce the problem of determining for a surface S, z(S), which is the maximum cochromatic number among all graphs G that embed in S. Some general bounds are obtained; for example, it is shown that if S is orientable of genus at least one, or if S is nonorientable of genus at least four, then z(S) is nonorientable of genus at least four, then z(S)≤χ(S). Here χ(S) denotes the chromatic number S. Exact results are obtained for the sphere, the Klein bottle, and for S. It is conjectured that z(S) is equal to the maximum n for which the graph G_n = K₁ ∪ K₂ ∪ … ∪ K_n embeds in S.     

Total restrained domination numbers of trees

For a given connected graph G=(V,E), a set D_tr⊆V(G) is a total restrained dominating set if it is dominating and both 〈D_tr〉 and 〈V(G)-D_tr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by γ_tr(G). In this paper we characterize the trees with equal total and total restrained dominating numbers and give a lower bound on the total restrained dominating number of a tree T in terms of its order and the number of leaves of T.     

Circular-arc digraphs: A characterization

Given a family F of ordered pairs of sets {(S_u,T_u): u ? V}, the intersection digraph of F is the digraph with vertices V defined by uv ? E(D) if S_u ≠ T_v Ø. Interval digraphs are intersection digraphs of families in which every set is an interval, and were characterized in a previous paper by conditions on the adjacency matrix. In this paper, another such condition leads to an adjacency matrix characterization of circular-arc digraphs, the intersection digraphs of families in which every set is an arc of a circle.     

Long dominating cycles and paths in graphs with large neighborhood unions

Let G be a graph of order n and define NC(G) = min{|N(u) ∪ N(v)| |uv ? E(G)}. A cycle C of G is called a dominating cycle or D-cycle if V(G) - V(C) is an independent set. A D-path is defined analogously. The following result is proved: if G is 2-connected and contains a D-cycle, then G contains a D-cycle of length at least min{n, 2NC(G)} unless G is the Petersen graph. By combining this result with a known sufficient condition for the existence of a D-cycle, a common generalization of Ore's Theorem and several recent “neighborhood union results” is obtained. An analogous result on long D-paths is also established.     

Pseudodifferential Operators on Prehomogeneous Vector Spaces

ABSTRACT

Let G be a connected, linear algebraic group defined over ?, acting regularly on a finite dimensional vector space V over ? with ?-structure V _?. Assume that V possesses a Zariski-dense orbit, so that (G, ?, V) becomes a prehomogeneous vector space over ?. We consider the left regular representation π of the group of ?-rational points G _? on the Banach space C₀(V _?) of continuous functions on V _? vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G _?. Denote the complement of the dense orbit by S, and put S _? = S ∩ V _?. It turns out that, on V _? ? S _?, π(f) is a smooth operator. If S _? = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V _? ? {0}. Its nonunique extension to V _? can then be regarded as a trace of π(f). If G is reductive, and S and S _? are irreducible hypersurfaces, π(f) corresponds, on each connected component of V _? ? S _?, to a totally characteristic pseudodifferential operator. In this case, the restriction of the Schwartz kernel of π(f) to the diagonal defines a distribution on V _? ? S _? given by some power |p(m)| ^s of a relative invariant p(m) of (G, ?, V) and, as a consequence of the Fundamental Theorem of Prehomogeneous Vector Spaces, its extension to V _?, and the complex s-plane, satisfies functional equations similar to those for local zeta functions. A trace of π(f) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.     

Linear preservers of orthogonal decomposable Tensors

Let G be a subgroup of the symmetric group S_m and V be an n-dimensional unitary space where nm. Let V(G) be the symmetry class of tensors over V associated with G and the identity character. Let D(G) be the set of all decomposable elements of V(G) and O(G) be its subset consisting of all nonzero decomposable tensors x ₁ ^?…^? x_m such that {x ₁,…,x_m } is an orthogonal set. In this paper we study the structure of linear mappings on V(G) that preserve one of the following subsets: (i)O(G), (ii) D(G)\(O(G)?{0}).     

Spanning trees with leaf distance at least four

For a graph G, we denote by i(G) the number of isolated vertices of G. We prove that for a connected graph G of order at least five, if i(G–S) < |S| for all ?? ≠ S ? V(G), then G has a spanning tree T such that the distance in T between any two leaves of T is at least four. This result was conjectured by Kaneko in “Spanning trees with constrains on the leaf degree”, Discrete Applied Math, 115 (2001), 73–76. Moreover, the condition in the result is sharp in a sense that the condition i(G–S) < |S| cannot be replaced by i(G–S) ≤ |S|. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 83–90, 2007     

On hamiltonian-connected graphs

One of the most fundamental results concerning paths in graphs is due to Ore: In a graph G, if deg x + deg y ≧ |V(G)| + 1 for all pairs of nonadjacent vertices x, y ? V(G), then G is hamiltonian-connected. We generalize this result using set degrees. That is, for S ? V(G), let deg S = |_x?S N(x)|, where N(x) = {v|xv ? E(G)} is the neighborhood of x. In particular we show: In a 3-connected graph G, if deg S₁ + deg S₂ ≧ |V(G)| + 1 for each pair of distinct 2-sets of vertices S₁, S₂ ? V(G), then G is hamiltonian-connected. Several corollaries and related results are also discussed.     

An extension of Vizing's theorem

Let G = (V (G), E (G)) be a simple graph of maximum degree δ ≤ D such that the graph induced by vertices of degree D is either a null graph or is empty. We give an upper bound on the number of colours needed to colour a subset S of V (G) ∪ E (G) such that no adjacent or incident elements of S receive the same colour. In particular, if S = E (G), we have the chromatic index χ′(G) ≤ D whereas if S = V (G) ∪ E (G) and for some positive integer k, we have the total chromatic number _χT(G) ≤ D + k. © 1997 John Wiley & Sons, Inc.     

A new domination conception

Let f be an integer valued function defined on the vertex set V(G) of a simple graph G. We call a subset D_f of V(G) a f-dominating set of G if |N(x, G) ∩ D_f| ≥ f(x) for all x ∈ V(G) — D_f, where N(x, G) is the set of neighbors of x. D_f is a minimum f-dominating set if G has no f-dominating set D′_f with |D_f| < |D_f|. If j, k ∈ N₀ = {0,1,2,…} with j ≤ k, then we define the integer valued function f_j,k on V(G) by . By μ_j,k(G) we denote the cardinality of a minimum f_j,k-dominating set of G. A set D ? V(G) is j-dominating if every vertex, which is not in D, is adjacent to at least j vertices of D. The j-domination number γ_j(G) is the minimum order of a j-dominating set in G. In this paper we shall give estimations of the new domination number μ_j,k(G), and with the help of these estimations we prove some new and some known upper bounds for the j-domination number. © 1993 John Wiley & Sons, Inc.     

The geodetic numbers of graphs and digraphs

For every two vertices u and v in a graph G,a u-v geodesic is a shortest path between u and v.Let I(u,v)denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I(S) denote the union of all I(u,v)for u,v∈S.The geodetic number g(G)of a graph G is the minimum cardinality of a set S with I(S)=V(G).For a digraph D,there is analogous terminology for the geodetic number g(D).The geodetic spectrum of a graph G,denoted by S(G),is the set of geodetic numbers of all orientations of graph G.The lower geodetic number is g~-(G)=minS(G)and the upper geodetic number is g~ (G)=maxS(G).The main purpose of this paper is to study the relations among g(G),g~-(G)and g~ (G)for connected graphs G.In addition,a sufficient and necessary condition for the equality of g(G)and g(G×K_2)is presented,which improves a result of Chartrand,Harary and Zhang.