3-Factor-criticality in domination critical graphs

A graph G is said to be k-γ-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k-1 vertices. The structure of k-γ-critical graphs remains far from completely understood when k?3.A graph G is factor-critical if G-v has a perfect matching for every vertex v∈V(G) and is bicritical if G-u-v has a perfect matching for every pair of distinct vertices u,v∈V(G). More generally, a graph is said to be k-factor-critical if G-S has a perfect matching for every set S of k vertices in G. In three previous papers [N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs, Discrete Math. 272 (2003) 5-15; N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs. II. Utilitas Math. 70 (2006) 11-32], we explored the toughness of 3-γ-critical graphs and some of their matching properties. In particular, we obtained some properties which are sufficient for a 3-γ-critical graph to be factor-critical and, respectively, bicritical. In the present work, we obtain similar results for k-factor-critical graphs when k=3.     

Generalization of matching extensions in graphs (III)

Proposing them as a general framework, Liu and Yu (2001) [6] introduced (n,k,d)-graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let n,k and d be non-negative integers such that n+2k+d+2?|V(G)| and |V(G)|−n−d is even. If on deleting any n vertices from G the remaining subgraph H of G contains a k-matching and each k-matching can be extended to a defect-d matching in H, then G is called an (n,k,d)-graph. In this paper, we obtain more properties of (n,k,d)-graphs, in particular the recursive relations of (n,k,d)-graphs for distinct parameters n,k and d. Moreover, we provide a characterization for maximal non-(n,k,d)-graphs.     

An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

Edge fault tolerance analysis of super k-restricted connected networks

An edge cut X of a connected graph G is a k-restricted edge cut if G-X is disconnected and every component of G-X has at least k vertices. Additionally, if the deletion of a minimum k-restricted edge cut isolates a connected component of k vertices, then the graph is said to be super-λ_k. In this paper, several sufficient conditions yielding super-λ_k graphs are given in terms of the girth and the diameter.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

Variations of Y-dominating functions on graphs

Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where N_G[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.     

Partitioning graphs into complete and empty graphs

Given integers j and k and a graph G, we consider partitions of the vertex set of G into j+k parts where j of these parts induce empty graphs and the remaining k induce cliques. If such a partition exists, we say G is a (j,k)-graph. For a fixed j and k we consider the maximum order n where every graph of order n is a (j,k)-graph. The split-chromatic number of G is the minimum j where G is a (j,j)-graph. Further, the cochromatic number is the minimum j+k where G is a (j,k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

The multiple originator broadcasting problem in graphs

Given a graph G and a vertex subset S of V(G), the broadcasting time with respect toS, denoted by b(G,S), is the minimum broadcasting time when using S as the broadcasting set. And the k-broadcasting number, denoted by b_k(G), is defined by b_k(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph G and two vertex subsets S, S^′ of V(G), define , d(S,S^′)=min{d(u,v)|u∈S, v∈S^′}, and for all v∈V(G). For all k, 1?k?|V(G)|, the k-radius of G, denoted by r_k(G), is defined as r_k(G)=min{d(G,S)|S⊆V(G), |S|=k}.In this paper, we study the relation between the k-radius and the k-broadcasting numbers of graphs. We also give the 2-radius and the 2-broadcasting numbers of the grid graphs, and the k-broadcasting numbers of the complete n-partite graphs and the hypercubes.     

Labeling trees with a condition at distance two

An L(h,k)-labeling of a graph G is an integer labeling of vertices of G, such that adjacent vertices have labels which differ by at least h, and vertices at distance two have labels which differ by at least k. The span of an L(h,k)-labeling is the difference between the largest and the smallest label. We investigate L(h,k)-labelings of trees of maximum degree Δ, seeking those with small span. Given Δ, h and k, span λ is optimal for the class of trees of maximum degree Δ, if λ is the smallest integer such that every tree of maximum degree Δ has an L(h,k)-labeling with span at most λ. For all parameters Δ,h,k, such that h<k, we construct L(h,k)-labelings with optimal span. We also establish optimal span of L(h,k)-labelings for stars of arbitrary degree and all values of h and k.     

Graphs with the maximum or minimum number of 1-factors

Recently Alon and Friedland have shown that graphs which are the union of complete regular bipartite graphs have the maximum number of 1-factors over all graphs with the same degree sequence. We identify two families of graphs that have the maximum number of 1-factors over all graphs with the same number of vertices and edges: the almost regular graphs which are unions of complete regular bipartite graphs, and complete graphs with a matching removed. The first family is determined using the Alon and Friedland bound. For the second family, we show that a graph transformation which is known to increase network reliability also increases the number of 1-factors. In fact, more is true: this graph transformation increases the number of k-factors for all k≥1, and “in reverse” also shows that in general, threshold graphs have the fewest k-factors. We are then able to determine precisely which threshold graphs have the fewest 1-factors. We conjecture that the same graphs have the fewest k-factors for all k≥2 as well.     

Edge-deletable IM-extendable graphs with minimum number of edges

A graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. For a nonnegative integer k, a graph G is called a k-edge-deletable IM-extendable graph, if, for every F⊆E(G) with |F|=k, G−F is IM-extendable. In this paper, we characterize the k-edge-deletable IM-extendable graphs with minimum number of edges. We show that, for a positive integer k, if G is ak-edge-deletable IM-extendable graph on 2n vertices, then |E(G)|≥(k+2)n; furthermore, the equality holds if and only if either G≅K_k+2,k+2, or k=4r−2 for some integer r≥3 and G≅C₅[N_2r], where N_2r is the empty graph on 2r vertices and C₅[N_2r] is the graph obtained from C₅ by replacing each vertex with a graph isomorphic to N_2r.     

On island sequences of labelings with a condition at distance two

An L(2,1)-labeling of a graph G is a function f from the vertex set of G to the set of nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1, and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between the pair of vertices x,y. The lambda number of G, denoted λ(G), is the minimum range of labels used over all L(2,1)-labelings of G. An L(2,1)-labeling of G which achieves the range λ(G) is referred to as a λ-labeling. A hole of an L(2,1)-labeling is an unused integer within the range of integers used. The hole index of G, denoted ρ(G), is the minimum number of holes taken over all its λ-labelings. An island of a given λ-labeling of G with ρ(G) holes is a maximal set of consecutive integers used by the labeling. Georges and Mauro [J.P. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] inquired about the existence of a connected graph G with ρ(G)≥1 possessing two λ-labelings with different ordered sequences of island cardinalities. This paper provides an infinite family of such graphs together with their lambda numbers and hole indices. Key to our discussion is the determination of the path covering number of certain 2-sparse graphs, that is, graphs containing no pair of adjacent vertices of degree greater than 2.     

A survey of stratified domination in graphs

A graph G is 2-stratified if its vertex set is partitioned into two nonempty classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F in G rooted at v. In this paper, we survey recent results on the F-domination number for various 2-stratified graphs F.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Resolvable group divisible designs with block size four and general index

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hⁿ and general index λ. The necessary conditions for the existence of such a design are n≥4, and . These necessary conditions are shown to be sufficient for all λ≥2, with the definite exceptions of (λ,h,n)∈{(3,2,6)}∪{(2j+1,2,4):j≥1}. The known existence result for λ=1 is also improved.     

Graph designs for the eight-edge five-vertex graphs

The existence of graph designs for the two nonisomorphic graphs on five vertices and eight edges is determined in the case of index one, with three possible exceptions in total. It is established that for the unique graph with vertex sequence (3, 3, 3, 3, 4), a graph design of order n exists exactly when and n≠16, with the possible exception of n=48. For the unique graph with vertex sequence (2,3,3,4,4), a graph design of order n exists exactly when , with the possible exceptions of n∈{32,48}.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

On symmetric graphs of valency five

A graph X, with a subgroup G of the automorphism group of X, is said to be (G,s)-transitive, for some s≥1, if G is transitive on s-arcs but not on (s+1)-arcs, and s-transitive if it is -transitive. Let X be a connected (G,s)-transitive graph, and G_v the stabilizer of a vertex v∈V(X) in G. If X has valency 5 and G_v is solvable, Weiss [R.M. Weiss, An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979) 43-48] proved that s≤3, and in this paper we prove that G_v is isomorphic to the cyclic group Z₅, the dihedral group D₁₀ or the dihedral group D₂₀ for s=1, the Frobenius group F₂₀ or F₂₀×Z₂ for s=2, or F₂₀×Z₄ for s=3. Furthermore, it is shown that for a connected 1-transitive Cayley graph of valency 5 on a non-abelian simple group G, the automorphism group of is the semidirect product , where R(G) is the right regular representation of G and .