Graphs having distance-n domination number half their order

For any positive integer n and any graph G a set D of vertices of G is a distance-n dominating set, if every vertex vV(G)−D has exactly distance n to at least one vertex in D. The distance-n domination number γ_=n(G) is the smallest number of vertices in any distance-n dominating set. If G is a graph of order p and each vertex in G has distance n to at least one vertex in G, then the distance-n domination number has the upper bound p/2 as Ore's upper bound on the classical domination number. In this paper, a characterization is given for graphs having distance-n domination number equal to half their order, when the diameter is greater or equal 2n−1. With this result we confirm a conjecture of Boland, Haynes, and Lawson.     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Weighted domination in triangle-free graphs

A weighted graph (G,w) is a graph G together with a positive weight-function on its vertex set w : V(G)→R^>0. The weighted domination number γ_w(G) of (G,w) is the minimum weight w(D)=∑_vDw(v) of a set DV(G) such that every vertex xV(D)−D has a neighbor in D. If ∑_vV(G)w(v)=|V(G)|, then we speak of a normed weighted graph. Recently, we proved that

for normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

Least domination in a graph

The least domination number γ_L of a graph G is the minimum cardinality of a dominating set of G whose domination number is minimum. The least point covering number _L of G is the minimum cardinality of a total point cover (point cover including every isolated vertex of G) whose total point covering number is minimum. We prove a conjecture of Sampathkumar saying that in every connected graph of order n 2. We disprove another one saying that γ_{L L} in every graph but instead of it, we establish the best possible inequality . Finally, in relation with the minimum cardinality γ_t of a dominating set without isolated vertices (total dominating set), we prove that the ratio γ_L/γ_t can be in general arbitrarily large, but remains bounded by if we restrict ourselves to the class of trees.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

Circular permutation graph family with applications

In a circular permutation diagram, there are two sets of terminals on two concentric circles: C_in and C_out. Given a permutation Π = [π₁, π₂, …, π_n], terminal i on C_in and terminal π_i on C_out are connected by a wire. The intersection graph G_c of a circular permutation diagram D_c is called a circular permutation graph of a permutation Π corresponding to the diagram D_c. The set of all circular permutation graphs of a permutation Π is called the circular permutation graph family of permutation Π. In this paper, we propose the following: (1) an O(V + E) time algorithm to check if a labeled graph G = (V, E) is a labeled circular permutation graph. (2) An O(n log n + nt) time algorithm to find a maximum independent set of a family, where n = Π and t is the cardinality of the output. (Number t in the worst case is O(n). However, if Π is uniformly distributed (and independent from i), its expected value is O(√n).) (3) An O(min(δV_clog logV_c,V_clogV_c) + E_c) time algorithm for finding a maximum independent set of a circular permutation diagram D_c, where δ is the minimum degree of vertices in the intersection graph G_c = (V_c,E_c) of D_c. (4) An O(n log log n) time algorithm for finding a maximum clique and the chromatic number of a circular permutation diagram, where n is the number of wires in the diagram.     

The coordinate representation of a graph and n-universal graph of radius 1

Every graph can be represented as the intersection graph on a family of closed unit cubes in Euclidean space Eⁿ. Cube vertices have integer coordinates. The coordinate matrix, A(G)={v_nk} of a graph G is defined by the set of cube coordinates. The imbedded dimension of a graph, Bp(G), is a number of columns in matrix A(G) such that each of them has at least two distinct elements v_nk≠v_pk. We show that Bp(G)=cub(G) for some graphs, and Bp(G)n−2 for any graph G on n vertices. The coordinate matrix uses to obtain the graph U of radius 1 with 3ⁿ⁻² vertices that contains as an induced subgraph a copy of any graph on n vertices.     

Diameter vulnerability of GC graphs

Concern over fault tolerance in the design of interconnection networks has stimulated interest in finding large graphs with maximum degree Δ and diameter D such that the subgraphs obtained by deleting any set of s vertices have diameter at most D′, this value being close to D or even equal to it. This is the so-called (Δ,D,D′,s)-problem. The purpose of this work has been to study this problem for s=1 on some families of generalized compound graphs. These graphs were designed by Gómez (Ars Combin. 29-B (1990) 33) as a contribution to the (Δ,D)-problem, that is, to the construction of graphs having maximum degree Δ, diameter D and an order large enough. When approaching the mentioned problem in these graphs, we realized that each of them could be redefined as a compound graph, the main graph being the underlying graph of a certain iterated line digraph. In fact, this new characterization has been the key point to prove in a suitable way that the graphs belonging to these families are solutions to the (Δ,D,D+1,1)-problem.     

The Hoffman number of a graph

For a connected graph G with n vertices, let {λ₁,λ₂,…,λ_r} be the set of distinct positive eigenvalues of the Laplacian matrix of G. The Hoffman number μ(G) of G is defined by μ(G)=λ₁λ₂…λ_r/n. In this paper, we study some properties and applications of the Hoffman number.     

Large eigenvalues of the laplacian

Let G be a simple graph on n vertices and let L=L(G) be the Laplacian matrix of G corresponding to some ordering of the vertices. It is known that λ≤n for any eigenvalue λ of L. In this note we characterize when n is an eigenvalue of L with multiplicity m.     

Rank, term rank, and chromatic number

Let G be a graph with a nonempty edge set, and with rank rk(G), term rank Rk(G), and chromatic number χ(G). We characterize Rk(G) as being the maximum number of colors in certain proper colorings of G. In particular, we observe that χ(G)Rk(G), with equality holding if and only if (besides isolated vertices) G is either complete or a star. For a twin-free graph G, we observe the bound and we show that this bound is sharp.     

No-hole L(2,1)-colorings

An L(2,1)-coloring of a graph G is a coloring of G's vertices with integers in {0,1,…,k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…,k}.

Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.     

The integrity of a cubic graph

Integrity, a measure of network reliability, is defined as

where G is a graph with vertex set V and m(G−S) denotes the order of the largest component of G−S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices:

Moreover, there exist an infinite family of connected cubic graphs whose integrity satisfies a linear lower bound I(G)>βn for some constant β. We provide a value for β, but it is likely not best possible. To prove the upper bound we first solve the following extremal problem. What is the least number of vertices in a cubic graph whose removal results in an acyclic graph? The solution (with a few minor exceptions) is that n/3 vertices suffice and this is best possible.     

On a problem concerning ordered colourings

Let G be a connected graph with v(G) 2 vertices and independence number (G). G is critical if for any edge e of G:

1. (i) (G − e) > (G), if e is not a cut edge of G, and

2. (ii) v(G_i) − (G_i) < v(G) − (G), I = 1, 2, if e is a cut edge and G₁, G₂ are the two components of G − e.

Recently, Katchalski et al. (1995) conjectured that: if G is a connected critical graph, then with equality possible if and only if G is a tree. In this paper we establish this conjecture.     

Cyclic coloring of plane graphs

Let G be a plane graph, and let χ_k(G) be the minimum number of colors to color the vertices of G so that every two of them which lie in the boundary of the same face of the size at most k, receive different colors. In 1966, Ore and Plummer proved that χ_k(G)2k for any k3. It is also known that χ₃(G)4 (Appel and Haken, 1976) and χ₄(G)6 (Borodin, 1984). The result in the present paper is: χ₅(G)9, χ₆(G)11, χ₇(G)12, and χ_k(G)2k − 3 if k8.     

Vertices contained in all or in no minimum total dominating set of a tree

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. We characterize the set of vertices of a tree that are contained in all, or in no, minimum total dominating sets of the tree.     

Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

A number of results in hamiltonian graph theory are of the form “ implies ”, where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. An example of such a theorem is the well-known Chvátal–Erd s Theorem, which states that every graph G with κ is hamiltonian. Here κ is the vertex connectivity of G and is the cardinality of a largest set of independent vertices of G. In another paper Chvátal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than κ independent vertices. In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of a well-known theorem of Jung (Ann. Discrete Math. 3 (1978) 129) for graphs on 16 or more vertices.     

k-Subdomination in graphs

For a positive integer k, a k-subdominating function of a graph G=(V,E) is a function f : V→{−1,1} such that ∑_uNG[v]f(u)1 for at least k vertices v of G. The k-subdomination number of G, denoted by γ_ks(G), is the minimum of ∑_vVf(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We also give a lower bound for γ_ks(G) in terms of the degree sequence of G. This generalizes some known results on the k-subdomination number γ_ks(G), the signed domination number γ_s(G) and the majority domination number γ_maj(G).     

Cell rotation graphs of strongly connected orientations of plane graphs with an application

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T⁻, thus (T,T⁻) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T⁻, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.