Gary Chartrand

Resolvability in graphs and the metric dimension of a graph

Abstract For an ordered subset W={w1,w2,…,wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v | W)=(d(v,w 1 ) , d(v,w2),…,d(v,wk)). The set W is a resolving set for G if r(u | W)=r(v | W) implies that u=v for all pairs u,v of vertices of G. The metric dimension dim(G) of G is the minimum cardinality of a resolving set for G. Bounds on dim(G) are presented in terms of the order and the diameter of G. All connected graphs of order n having dimension 1,n−2, or n−1 are determined. A new proof for the dimension of a tree is also presented. From this result sharp bounds on the metric dimension of unicyclic graphs are established. It is shown that dim(H)⩽dim(H×K2)⩽dim(H)+1 for every connected graph H. Moreover, it is shown that for every positive real number e, there exists a connected graph G and a connected induced subgraph H of G such that dim(G)/dim(H)

On the Geodetic Number of a Graph

For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. The geodetic number g(G) is the minimum cardinality among the subsets S of V(G) with I(S) = V(G). It is shown that if G is a graph of order n and diameter d then g(G) ≤ n − d + 1 and this bound is sharp. For positive integers r, d, and k ≥ 2 with r ≤ d ≤ 2r, there exists a connected graph G of radius r, diameter d, and g(G) = k. Also, for integers n, d, and k with 2 ≤ d < n, 2 ≤ k < n, and n − d − k + 1 ≥ 0, there exists a graph G of order n, diameter d, and g(G) = k. It is shown, for every nontrivial connected graph G, that g(G) ≤ g(G × K2). A sufficient condition for the equality of g(G) and g(G × K2) is presented. A graph F is a minimum geodetic subgraph if there exists a graph G containing F as an induced subgraph such that V(F) is a minimum geodetic set for G. Minimum geodetic subgraphs are characterized.

Resolvability and the upper dimension of graphs

Abstract For an ordered set W = { w 1 , w 2 ,…, w k } of vertices and a vertex v in a connected graph G , the (metric) representation of v with respect to W is the k -vector r ( v | W ) = ( d ( v , w 1 ), d ( v , w 2 ),…, d ( v , w k )), where d ( x , y ) represents the distance between the vertices x and y . The set W is a resolving set for G if distinct vertices of G have distinct representations. A new sharp lower bound for the dimension of a graph G in terms of its maximum degree is presented. A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim( G ). A resolving set S of G is a minimal resolving set if no proper subset of S is a resolving set. The maximum cardinality of a minimal resolving set is the upper dimension dim + ( G ). The resolving number res( G ) of a connected graph G is the minimum k such that every k -set W of vertices of G is also a resolving set of G . Then 1 ≤ dim( G ) ≤ dim + ( G ) ≤ res( G ) ≤ n − 1 for every nontrivial connected graph G of order n . It is shown that dim + ( G ) = res( G ) = n − 1 if and only if G = K n , while dim + ( G ) = res( G ) = 2 if and only if G is a path of order at least 4 or an odd cycle. The resolving numbers and upper dimensions of some well-known graphs are determined. It is shown that for every pair a , b of integers with 2 ≤ a ≤ b , there exists a connected graph G with dim( G ) = dim + ( G ) = a and res( G ) = b . Also, for every positive integer N , there exists a connected graph G with res( G ) − dim + ( G ) ≥ N and dim + ( G ) − dim( G ) ≥ N .

The forcing geodetic number of a graph

For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. A set S is a geodetic set if I(S) = V (G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T . The forcing geodetic number fG(S) of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a, b) / ∈ {(1, 1), (2, 2)}.

On Uniquely Colorable Planar Graphs

Abstract A labeled graph G with chromatic number n is called uniquely n -colorable or simply uniquely colorable if every two partitions of the point set of G into n color classes are the same. Uniquely colorable planar graphs are investigated. In particular, it is shown that uniquely 3-colorable planar graphs with at least four points contain at least two triangles, uniquely 4-colorable planar graphs are maximal planar, and uniquely 5-colorable planar graphs do not exist.

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.

The Steiner number of a graph

Abstract For a connected graph G of order n⩾3 and a set W⊆V(G) , a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W⊆V(T) . The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W . The set W is a Steiner set for G if S(W)=V(G) . The minimum cardinality among the Steiner sets of G is the Steiner number s(G) . Connected graphs of order n with Steiner number n , n−1 , or 2 are characterized. It is shown that every pair k,n of integers with 2⩽k⩽n is realizable as the Steiner number and order of some connected graph. For positive integers r,d, and k⩾2 with r⩽d⩽2r , there exists a connected graph of radius r , diameter d , and Steiner number k . Also, for integers n,d, and k with 2⩽d , 2⩽k , and n−d−k+1⩾0 , there exists a graph G of order n , diameter d , and Steiner number k . For two vertices u and v of a connected graph G , the set I[u,v] consists of all vertices lying on some u – v geodesic in G . For U⊆V(G) , the set I[U] is the union of all sets I[u,v] for u,v∈U . A set U is a geodetic set if I[U]=V(G) . The cardinality of a minimum geodetic set is the geodetic number g(G) . It is shown that g(G)⩽s(G) and that for every two integers a and b such that 3⩽a⩽b , there exists a graph G of radius r and diameter d such that d=r+1 , g(G)=a , and s(G)=b .

Geodetic sets in graphs

For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u − v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v ∈ S. If I[S] = V (G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for every two distinct vertices u, v ∈ S, there exists a third vertex w of G that lies in some u− v geodesic but in no x− y geodesic for x, y ∈ S and {x, y} 6= {u, v}. It is shown that for every integer k ≥ 2, there exists a connected graph G with g(G) = k which contains a uniform, essential minimum geodetic set. A minimal geodetic set S has no proper subset which is a geodetic set. The maximum cardinality of a minimal geodetic set is the upper geodetic number g(G). It is shown that every two integers a and b with 2 ≤ a ≤ b are realizable as the geodetic and upper geodetic numbers, respectively, of some graph and when a < b the minimum order of such a graph is b + 2.

The Geodetic Number of an Oriented Graph

For two vertices u and v of an oriented graph D, the set I(u, v) consists of all vertices lying on a u?v geodesic or v?u geodesic in D. If S is a set of vertices of D, then I(S) is the union of all sets I(u, v) for vertices u and v in S. The geodetic number g(D) is the minimum cardinality among the subsets S of V(D) with I(S) =V(D). Several results concerning the geodetic numbers of connected oriented graphs are presented. For a nontrivial connected graph G, the lower orientable geodetic number g?(G) of G is the minimum geodetic number among the orientations of G and the upper orientable geodetic number g+(G) is the maximum such geodetic number. It is shown that g?(G)?g+(G) for every connected graph of order at least 3. The lower and upper orientable geodetic numbers of several well known classes of graphs are determined. It is shown that for every two integers n and m with 1 ?n? 1 ?m? (n 2) , there exists a connected graph G of order n and size m such that g+(G) =n.

Radio k-colorings of paths

For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that d(u, v) + |c(u)− c(v)| ≥ 1 + k for every two distinct vertices u and v of G, where d(u, v) is the distance between u and v. The value rck(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rck(G) of G is the minimum value of rck(c) taken over all radio k-colorings c of G. In this paper, radio k-colorings of paths are studied. For the path Pn of order n ≥ 9 and n odd, a new improved bound for rcn−2(Pn) is presented. For n ≥ 4, it is shown that rcn−3(Pn) ≤ ∗Research supported in part by the Western Michigan University Arts and Sciences Teaching and Research Award Program. 6 G. Chartrand, L. Nebeský and P. Zhang ( n−2 2 ) + 2. Upper and lower bounds are also presented for rck(Pn) in terms of k when 1 ≤ k ≤ n−1. The upper bound is shown to be sharp when 1 ≤ k ≤ 4 and n is sufficiently large.