David Erwin

Broadcasts in graphs

We say that a function f : V → {0, 1, ..., diam(G)} is a broadcast if for every vertex v ∈ V, f(v) ≤ e(v), where diam(G) denotes the diameter of G and e(v) denotes the eccentricity of v. The cost of a broadcast is the value f(V) = Σv∈V f(v). In this paper we introduce and study the minimum and maximum costs of several types of broadcasts in graphs, including dominating, independent and efficient broadcasts.

Boundary vertices in graphs

The distance d(u, v) between two vertices u and v in a nontrivial connected graph G is the length of a shortest u-v path in G. For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. A vertex v of G is a peripheral vertex if e(v) is the diameter of G. The subgraph of G induced by its peripheral vertices is the periphery Per(G) of G. A vertex u of G is an eccentric vertex of a vertex v if d(u, v)= e(v). A vertex x is an eccentric vertex of G if x is an eccentric vertex of some vertex of G. The subgraph of G induced by its eccentric vertices is the eccentric subgraph Ecc(G) of G. A vertex u of G is a boundary vertex of a vertex v if d(w,v) ≤ d(u,v) for all w ∈ N(u). A vertex u is a boundary vertex of G if u is a boundary vertex of some vertex of G. The subgraph of G induced by its boundary vertices is the boundary ∂(G) of G. A graph H is a boundary graph if H = ∂(G) for some graph G. We study the relationship among the periphery, eccentric subgraph, and boundary of a connected graph and establish a characterization of all boundary graphs. It is shown that per each triple a, b, c of integers with 2 ≤ a ≤ b ≤ c, there is a connected graph G such that Per(G) has order a, Ecc(G) has order b, and ∂(G) has order c. Moreover, for each triple r,s,t of rational numbers with 0 < r ≤ s ≤ t ≤ 1, there is a connected graph G of order n such that |V(Per(G))|/n=r, |V(Ecc(G))|/n = s, and |V(∂(G))| n=t.

Graphs of order n with locating-chromatic number n - 1

For a coloring c of a connected graph G, let Π = (C1, C2, ..., Ck) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code cΠ(v) of v is the ordered k-tuple (d(v, C1), d(v, C2),..., d(v, Ck)), where d(v, Ci) = min{d(v, x): x ∈ Ci} for 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c is called a locating-coloring. The locating-chromatic number χL(G) is the minimum number of colors in a locating-coloring of G. It is shown that if G is a connected graph of order n ≥ 3 containing an induced complete multipartite subgraph of order n - 1, then (n + 1)/2 ≤ χL(G) ≤ n and, furthermore, for each integer k with (n + 1)/2 ≤ k ≤ n, there exists such a graph G of order n with χL(G) = k. Graphs of order n containing an induced complete multipartite subgraph of order n - 1 are used to characterize all connected graphs of order n ≥ 4 with locating-chromatic number n - 1.

The Decomposition Dimension of Graphs

Abstract. For an ordered k-decomposition ? = {G1, G2,…,Gk} of a connected graph G and an edge e of G, the ?-representation of e is the k-tuple r(e|?) = (d(e, G1), d(e, G2),…,d(e, Gk)), where d(e, Gi) is the distance from e to Gi. A decomposition ? is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n≥ 2, there exists a tree T of order n with dec(T) = k. It is also shown that dec(G) ≤n for every graph G of order n≥ 3 and that dec(Kn) ≤⌊(2n + 5)/3⌋ for n≥ 3.

On γ-labelings of trees

Let G be a graph of order n and size m. A γ-labeling of G is a oneto-one function f : V (G) → {0, 1, 2, . . . , m} that induces a labeling f ′ : E(G) → {1, 2, . . . ,m} of the edges of G defined by f ′(e) = |f(u)−f(v)| for each edge e = uv of G. The value of a γ-labeling f is val(f) = ∑ e∈E(G) f ′(e). The maximum value of a γ-labeling of G is defined as valmax(G) = max{val(f) : f is a γ-labeling of G}; while the minimum value of a γ-labeling of G is valmin(G) = min{val(f) : f is a γ-labeling of G}. 364 G. Chartrand, D. Erwin, D.W. VanderJagt and P. Zhang The values valmax(Sp,q) and valmin(Sp,q) are determined for double stars Sp,q. We present characterizations of connected graphs G of order n for which valmin(G) = n or valmin(G) = n + 1.

Orientation distance graphs

For two nonisomorphic orientations D and D2 of a graph G, the orientation distance do(D,D2) between D and D2 is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D2. The orientation distance graph 𝒟o(G) of G has the set 𝒪(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D2 of 𝒟0(G) are adjacent if and only if do(D,D2) = 1. For a nonempty subset S of 𝒪(G), the orientation distance graph 𝒟0(S) of S is the induced subgraph )S* of 𝒟o(G). A graph H is an orientation distance graph if there exists a graph G and a set S⊆ 𝒪(G) such that 𝒟o(S) is isomorphic to H. In this case, H is said to be an orientation distance graph with respect to G. This paper deals primarily with orientation distance graphs with respect to paths. For every integer n ≥4, it is shown that 𝒟o(Pn) is Hamiltonian if and only if n is even. Also, the orientation distance graph of a path of odd order is bipartite. Furthermore, every tree is an orientation distance graph with respect to some path, as is every cycle, and for n ≥ 3 the clique number of 𝒟o(Pn) is 2 if n is odd and is 3 otherwise.