The graphs in this note are finite and undirected . We allow multiple edges , but forbid loops . We shall use the notation of Bondy and Murty [1] , unless otherwise stated . Let G be a graph with E ( G ) ? [ . Let τ ( G ) denote the number of edge-disjoint spanning trees of G . For X ‘ E ( G ) , the notation G ( X ) denotes the spanning subgraph of G with edge set X , whereas G [ X ] denotes th...

Given an edge-weighted graph G = (V,E) and a set E0 ⊂ E, the incremental network design problem with minimum spanning trees asks for a sequence of edges e1, . . . , e ′ r ∈ E \ E0 minimizing ∑T t=1 w(Xt) where w(Xt) is the weight of a minimum spanning tree Xt for the subgraph (V,E0 ∪ {e1, . . . , et}) and T = |E \ E0|. We prove that this problem can be solved by a greedy algorithm.

Let Qd denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Qd is said to be (Qd, Qm)-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd) \E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ≥ 2 the minimum number of edges in a (Qd, Qm)-saturated graph is Θ(2 d). We also study weak satura...

Throughout, we shall understand a graph to be an undirected, connected graph without loops or multiple edges, and a spanning tree of a graph G to be a connected subgraph of G without circuits which includes every vertex of G. In [ 11, Sedlacek has asked which graphs are uniquely reconstructable from their spanning trees, and calls such graphs URST graphs. More precisely, we say G is a URST grap...

A graph is called 2K2-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every 3 2 -tough 2K2-free graph with at least three vertices has a spanning trail with maximum degree at most 4. In this paper, we confirm this conjecture. We also provide examples for all t < 54 of t-tough graphs that do not have a spanning trail with maximum degre...

A Halin graph is a plane graph constructed from a planar drawing of a tree by connecting all leaves of the tree with a cycle which passes around the boundary of the graph. The tree must have four or more vertices and no vertices of degree two. Halin graphs have many nice properties such as being Hamiltonian and remain Hamiltonian after any single vertex deletion. In 1975, Lovász and Plummer con...

Given an undirected graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of pairs (F, S) where S is a nonempty subset of nodes and F is a subset of the edges with both endnodes in S. We obtain a strong relationship between the non-empty subgraph polytope and the spanning forest polytope. We further show that these polytopes provide polynomial size extended for...

Let Φ be an embedding of graph G in a surface S. If there exists a subset K of S bounded by a subgraph of G which contains all the vertices of G, then K is called a spanning subset of Φ. Examples of spanning subsets include spanning discs, spanning annuli with some number of holes (called planarizing sets in some papers). A spanning subset may provide a simpler structure but still contain enoug...

A minimal detour subgraph of the n-dimensional cube is a spanning subgraph G of Qn having the property that for vertices x, y of Qn, distances are related by dG(x; y) dQn(x; y) + 2. For a spanning subgraph G of Qn to be a local detour subgraph, we require only that the above inequality be satis ed whenever x and y are adjacent in Qn. Let f(n) (respectively, fl(n) ) denote the minimum number of ...

A spanning tree of a graph G is a connected acyclic spanning subgraph of G. We consider enumeration of spanning trees when G is a 2-tree, meaning that G is obtained from one edge by iteratively adding a vertex whose neighborhood consists of two adjacent vertices. We use this construction order both to inductively list the spanning trees without repetition and to give bounds on the number of the...