https://ogma.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:20800 Wed 27 Jul 2022 13:49:56 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:14455 Wed 11 Apr 2018 13:25:33 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19330 Thu 06 Aug 2015 10:47:12 AEST ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:7847 Sat 24 Mar 2018 10:48:16 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13205 ir(G) (open irratic number X_oir(G)of a graph G equals the minimum order of an irredundant (open irredundant) coloring of G. In this paper we present several results on these two parameters.]]> Sat 24 Mar 2018 10:35:49 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13454 Xd(G). In the DOMINATOR COLORING (DC) problem, a graph G and a positive integer k are given as input and the objective is to check whether _Xd(G) ≤ k. We first show that unless P=NP, DC cannot be solved in polynomial time on bipartite, planar, or split graphs. This resolves an open problem posed by Chellali and Maffray [Dominator Colorings in Some Classes of Graphs, Graphs and Combinatorics, 2011] about the polynomial time solvability of DC on chordal graphs. We then complement these hardness results by showing that the problem is fixed parameter tractable (FPT) on chordal graphs and in graphs which exclude a fixed apex graph as a minor.]]> Sat 24 Mar 2018 08:17:07 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:13453 k = {0, 1, 2, …, k – 1} be the set of integers modulo k. Let D_k(x,y) = min{|x – y|,k – |x – y|} for x,y ∈ Z_k. A pseudo complete d-coloring of G using k colors is a mapping ϕ : V(G) → Z_k such that for any two elements i, j ∈ Z_k with D_k (i,j) ≥ d, there exist adjacent vertices u, v such that ϕ(u) = i and ϕ(v) = j. The maximum value of k for which G is k-pseudo complete d-colorable is called the pseudo d-achromatic number of G and is denoted by ψ^d_s(G). A graph G is called k-edge d-critical if ψ^d_s(G) = k and ψ^d_s(G - e) < k for all e ∈ E(G). In this paper we present several basic results on the degrees and degree sequence of k-edge d-critical graphs.]]> Sat 24 Mar 2018 08:17:07 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:21683 G = (V, E) be a graph. A subset S of V is called an equivalence set if every component of the induced subgraph <S is complete. In this paper starting with the concept of equivalence set as seed property, we form an inequality chain of six parameters, which we call the equivalence chain of G. WE present several basic results on these parameters and problems for further investigation.]]> Sat 24 Mar 2018 08:02:55 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:21607 a, d)-edge-antimagic total labeling of a graph G with p vertices and q edges is a bijection f from the set of all vertices and edges to the set of positive integers {1, 2, 3, . . . , p + q} such that all the edge-weights w(uv) = f(u) + f(v) + f(uv); uv ∈ E(G), form an arithmetic progression starting from a and having common difference d. An (a, d)-edge-antimagic total labeling is called a super (a, d)-edge-antimagic total labeling ((a, d)-SEAMT labeling) if f(V (G)) = {1, 2, 3, . . . , p}. In this paper we investigate the existence of super (a, d)-edge antimagic total labeling for friendship graphs and generalized friendship graphs.]]> Sat 24 Mar 2018 07:59:31 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:21856 1,...,w_k) be an ordered subset of V. The k-vector r(v|W)=(d(v,w₁),...,d(v,w_k)) is called the metric representation of v with respect to W. The set W is a resolving set of G if r(u|W) = r(v|W) implies u = v. The minimum cardinality of a resolving set in G is the metric dimension of G. In this paper we extend the notion of metric dimension to hypergraphs. We also introduce the dual concept, that is, edge dimension for hypergraphs, and initiate a study on this parameter.]]> Sat 24 Mar 2018 07:59:12 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:21172 Sat 24 Mar 2018 07:58:04 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:20696 D(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, Σ_u∈ND(v) f(u) is a constant, called D-vertex magic constant. O'Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.]]> Sat 24 Mar 2018 07:55:40 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19331 Sat 24 Mar 2018 07:52:14 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:19337 S of the vertex set of a hypergraph ℋ is called a dominating set of ℋ if for every vertex v not in S there exists u ∈ S such that u and v are contained in an edge in ℋ. The minimum cardinality of a dominating set in ℋ is called the domination number of ℋ and is denoted by γ(ℋ). A transversal of a hypergraph ℋ is defined to be a subset T of the vertex set such that T ⋂ E ≠ Ø for every edge E of ℋ. The transversal number of ℋ, denoted by t.(ℋ), is the minimum number of vertices in a transversal. A hypergraph is of rank k if each of its edges contains at most k vertices. The inequality t(ℋ) = γ(ℋ) is valid for every hypergraph ℋ without isolated vertices. In this paper, we investigate the hypergraphs satisfying t(ℋ) = γ(ℋ), and prove that their recognition problem is NP-hard already on the class of linear hypergraphs of rank 3, while on unrestricted problem instances it lies inside the complexity class ϴ ^p₂. Structurally we focus our attention on hypergraphs in which each subhypergraph ℋ¹ without isolated vertices fulfills the equality t(ℋ¹) = (ℋ¹). We show that if each induced subhypergraph satisfies the equality then it holds for the non-induced ones as well. Moreover, we prove that for every positive integer k, there are only a finite number of forbidden subhypergraphs of rank k, and each of them has domination number at most k.]]> Sat 24 Mar 2018 07:52:13 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27668 Sat 24 Mar 2018 07:38:51 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:29690 (a,d)-edge antimagic total labeling of a (p, q)-graph G is bijection f:V∪E→{1,2,3,…,p+q} with the property that the edge-weights w(uv)=f(u)+f(v)+f(uv) where uv∈E(G) form an arithmetic progression a,a+d,…,a+(q−1)d, where a > 0 and d ≥ 0 are two fixed integers. If such a labeling exists, then G is called an (a,d)-edge antimagic total graph. If further the vertex labels are the integers {1,2,3,…,p}, then f is called a super (a,d)-edge antimagic total labeling of G ((a, d)-SEAMT labeling) and a graph which admits such a labeling is called a super (a,d)-edge antimagic total graph ((a, d)-SEAMT graph). If d=0, then the graph G is called a super edge-magic graph. In this paper we investigate the existence of super (a, 3)-edge antimagic total labelings for union of two stars.]]> Sat 24 Mar 2018 07:38:47 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:28824 G = (V, E) be a graph without isolated vertices. Let m*(G) denote the optimal pixel expansion of a visual cryptography scheme for the strong access structure Γ(G) on V whose basis is E. In this paper we determine m*(G) for any connected graph G with vertex covering number two.]]> Sat 24 Mar 2018 07:38:27 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27687 Sat 24 Mar 2018 07:37:03 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27689 f={v ∈ V: f(v) > 0} is the positive set of f and is the boundary set of f. The relation P defined on the set F of all MTDFs of G by fpg if P_f = P_g and B_f = B_g is an equivalence relation which partitions F into a finite number of equivalence classes X₁, X₂,... X_t. The total convexity graph CT (G) of G has {X₁, X₂,...X_t} as its vertex set and X_i is adjacent to X_j if there exist f ∈ X_i and g ∈ X_j such that any convex combination of f and g is an MTDF of G. In this paper we determine the total convexity graphs of some standard graphs.]]> Sat 24 Mar 2018 07:37:03 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:27527 Sat 24 Mar 2018 07:28:58 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:26705 e(G) is the minimum cardinality of an equivalence dominating set of G. In this paper we investigate the structure of graphs G satisfying γ_e (G) = ∣V (G)∣ – ∆(G).]]> Sat 24 Mar 2018 07:26:20 AEDT ]]> https://ogma.newcastle.edu.au/vital/access/ /manager/Repository/uon:23752 f(G)=min{|g|:g is a minimal resolving function of G}, where |g|=∑v∈Vg(v). In this paper we study this parameter.]]> Sat 24 Mar 2018 07:11:10 AEDT ]]>