The domination number γ(G) of a graph G is the minimum number of vertices in a set D such that every vertex of the graph is either in D or is adjacent to a member of D. Any dominating set D of a graph G with |D| = γ(G) is called a γ-set of G. A vertex x of a graph G is called: (i) γ-good if x belongs to some γ-set and (ii) γ-bad if x belongs to no γ-set. The bondage number b(G) of a nonempty gr...

I study weakly efficient auctions with entry costs, under the IPV assumption, following Tan and Yilankaya [Tan, G., Yilankaya, O., 2006. Equilibria in second price auctions with participation costs. Journal of Economic Theory 130, 205–219]. First, I generalize their Proposition 4 to what I call (generalized) intuitive equilibrium. By such I prove that if bidders’ valuation distributions are ord...

Color is one of the fundamental elements in visual communication design. Various research has been conducted about color related. Thus, this was to find mapping related design using bibliometric analysis methods. The data used study contained Scopus database. Analysis based on information available website that collaborated with VOSViewer visualize results. results found there were 377 document...

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardina...

A Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number...

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

A subset, D, of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in D or adjacent to some vertex in D. The maximum cardinality of a partition of the vertex set of G into dominating sets is the domatic number of G, denoted d(G). G is said to be domatically critical if the removal of any edge of G decreases the domatic number, and G is domatically full if ...

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

A dominating setD ⊆ V of a graphG = (V,E) is said to be a connected cototal dominating set if 〈D〉 is connected and 〈V −D〉 6= ∅, contains no isolated vertices. A connected cototal dominating set is said to be minimal if no proper subset of D is connected cototal dominating set. The connected cototal domination number γccl(G) of G is the minimum cardinality of a minimal connected cototal dominati...

13 A contribution is made to the classification of lattice-like total perfect 14 codes in integer lattices Λn via pairs (G,Φ) formed by abelian groups G 15 and homomorphisms Φ : Z → G. A conjecture is posed that the cited 16 contribution covers all possible cases. A related conjecture on the unfinished 17 work on open problems on lattice-like perfect dominating sets in Λn with 18 induced compon...