A Halin graph is a simple plane graph consisting of a tree without degree 2 vertices and a cycle induced by the leaves of the tree. In 1975, Lovász and Plummer conjectured that every 4-connected plane triangulation has a spanning Halin subgraph. In this paper, we construct an infinite family of counterexamples to the conjecture.

In this paper we study the L(2, 1)-labeling problem on oriented planar graphs with particular attention on the subclasses of oriented prisms, Halin and cactus graphs. For these subclasses more accurate results are presented. keywords: L(2, 1)-labeling, oriented graph coloring, digraphs, prisms, Halin graphs, cacti.

In this paper we study the L(2, 1)-labeling problem on oriented planar graphs with particular attention on the subclasses of oriented prisms, Halin and cactus graphs. For these subclasses more accurate results are presented. keywords: L(2, 1)-labeling, oriented graph coloring, digraphs, prisms, Halin graphs, cacti.

We prove that the pathwidth of Halin graphs can be 3-approximated in linear time. Our approximation algorithms is based on a combinatorial result about respectful edge orderings of trees. Using this result we prove that the linear width of Halin graph is always at most three times the linear width of its skeleton. © 2005 Elsevier B.V. All rights reserved.

A tree T with no vertex of degree 2 is called a homeomorphically irreducible tree (HIT) and if T is spanning in a graph, then T is called a homeomorphically irreducible spanning tree (HIST). Albertson, Berman, Hutchinson and Thomassen asked if every triangulation of at least 4 vertices has a HIST and if every connected graph with each edge in at least two triangles contains a HIST. These two qu...

In this paper, we study how to draw Halin-graphs, i.e., planar graphs that consist of a tree $T$ and cycle among the leaves tree. Based on tree-drawing algorithms pathwidth $ pw(T) $, well-known graph parameter, find poly-line drawings height at most $6pw(T)+3\in O(\log n)$. We also give an algorithm for straight-line drawings, achieve $12pw(T)+1$ smaller if Halin-graph is cubic. show achieved ...

A Halin graph H is a planar graph obtained by drawing a tree T in the plane, where T has no vertex of degree 2, then drawing a cycle C through all leaves in the plane. We write H = T ∪ C, where T is called the characteristic tree and C is called the accompanying cycle. The problem is to find a spanning tree with the maximum number of leaves in a Halin graph. In this paper, we prove that the cha...

We describe and implement two local reduction rules that can be used to recognize Halin graphs in linear time, avoiding the complicated planarity testing step of previous linear time Halin graph recognition algorithms. The same two rules can be used as the basis for linear-time algorithms for other algorithmic problems on Halin graphs, including decomposing these graphs into a tree and a cycle,...

For any graph G , G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number ( ) k G of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number ( ) k G for a graph G and characterizing a graph by its competition number has been one of important research pr...

The Strong Chromatic Index of Halin Graphs By Ziyu Hu A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if they have adjacent endpoints. The strong chromatic index of a graph G, denoted by χs(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a planar graph constructed...