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...

For a graph G we define a graph T (G) whose vertices are the triangles in G and two vertices of T (G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k-connected graph and T (G) contains no edge then G admits a kcontractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabaya...

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic ce...

A directed graph G is an unavoidable subgraph o f all n-tournaments or, simply n-unavoidable, if every tournament on n vertices contain san isomorphic copy o f G, i.e., for each ntournament T there exists an edge preserving injection o f the vertices o f G into the vertices o f T. The problem o f showing certain types o f graphs to be n-unavoidable has been the subject o f several papers, for e...

For graphs G and H, an H-coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H, which connected n-vertex graph with minimum degree δ maximizes the number of H-colorings? We show that for non-regular H and sufficiently large n, the complete bipartite graph Kδ,n−δ is the unique maxim...

Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width k of a graph G is defined as the minimum integer d for which there exist at least k i...

A graph G is s-Hamiltonian if for any S ⊆ V (G) of order at most s, G−S has a Hamiltonian-cycle, and s-Hamiltonian connected if for any S ⊆ V (G) of order at most s, G − S is Hamiltonian-connected. Let k > 0, s ≥ 0 be two integers. The following are proved in this paper: (1) Let k ≥ s+ 2 and s ≤ n− 3. If G is a k-connected graph of order n and if max{d(v) : v ∈ I} ≥ (n+s)/2 for every independen...

Given a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings’ theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2-connected graphs, that are not...

Henning, A.M. and H.C. Swart, Bounds relating generalized domination parameters, Discrete Mathematics 120 (1993) 933105. The domination number r(G) and the total domination number y,(G) of a graph G are generalized to the K,-domination number 7x,(G) and the total K,-domination number y;_(G) for n>2, where y(G)=y,,(G) and Y~(G)=~~~(G), K,-connectivity is defined and, for every integer n >2, the ...

Let u and v be any two distinct vertices of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A wcontainer C(u, v) of G is a w∗-container if it contains all the vertices of G. A graph G is w∗-connected if there exists a w∗-container between any two distinct vertices. Let κ(G) be the connectivi...