Halin Graphs Research Articles

O-87 The role of CLEVER-1 in breast tumour metastasis

A piecewise linear simple curve in the plane made up of k+1 line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a k-bend path. Given a graph G, a collection of k-bend paths in which each path corresponds to a vertex in G and two paths have a common point if and only if the vertices corresponding to them are adjacent in G is called a Bk-VPG representation of G. Similarly, a collection of k-bend paths each of which corresponds to a vertex in G is called an Bk-EPG representation of G if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in G. The VPG bend-number bv(G) of a graph G is the minimum k such that G has a Bk-VPG representation. Similarly, the EPG bend-number be(G) of a graph G is the minimum k such that G has a Bk-EPG representation. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph then bv(G)≤1 and be(G)≤2. These bounds are tight. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths.

Labelling of some planar graphs with a condition at distance two

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale's channel assignment problem, which was first explored by Griggs and Yeh. For positive integer p ≥ q, the λp,q-number of graph G, denoted λ(G;p, q), is the smallest span among all integer labellings of V(G) such that vertices at distance two receive labels which differ by at least q and adjacent vertices receive labels which differ by at least p. Van den Heuvel and McGuinness have proved that λ(G;p, q) ≤ (4q-2)Δ+10p+38q-24 for any planar graph G with maximum degree Δ. In this paper, we studied the upper bound of λp,q-number of some planar graphs. It is proved that λ(G;p,q) ≤ (2q-1)Δ+2(2p-1) if G is an outerplanar graph and λ(G;p, q) ≤ (2q-1)Δ+6p-4q-1 if G is a Halin graph.

Minimum cycle bases of graphs on surfaces

In this paper we study the cycle base structures of embedded graphs on surfaces. We first give a sufficient and necessary condition for a set of facial cycles to be contained in a minimum cycle base (or MCB in short) and then set up a 1–1 correspondence between the set of MCBs and the set of collections of nonseparating cycles which are in general positions on surfaces and are of shortest total length. This provides a way to enumerate MCBs in a graph via nonseparating cycles. In particular, some known results such as P.F. Stadler's work on Halin graphs [Minimum cycle bases of Halin graphs, J. Graph Theory 43 (2003) 150–155] and Leydold and Stadler's results on outer-planar graphs [Minimum cycle bases of outerplanar graphs, Electronic J. Combin. 5(16) (1998) 14] are concluded. As applications, the number of MCBs in some types of graphs embedded in lower surfaces (with arbitrarily high genera) is found. Finally, we present an interpolation theorem for the number of one-sided cycles contained in MCB of an embedded graph.

The 2-dipath chromatic number of Halin graphs

A 2 -dipath k-coloring f of an oriented graph G → is a mapping from V ( G → ) to the color set { 1 , 2 , … , k } such that f ( x ) ≠ f ( y ) whenever two vertices x and y are linked by a directed path of length 1 or 2. The 2 -dipath chromatic number χ → 2 ( G → ) of G → is the smallest k such that G → has a 2-dipath k-coloring. In this paper we prove that if G → is an oriented Halin graph, then χ → 2 ( G → ) ⩽ 7 . There exist infinitely many oriented Halin graphs G → such that χ → 2 ( G → ) = 7 .

On the oriented chromatic number of Halin graphs

An oriented k-coloring of an oriented graph G is a mapping c : V ( G ) → { 1 , 2 , … , k } such that (i) if x y ∈ E ( G ) then c ( x ) ≠ c ( y ) and (ii) if x y , z t ∈ E ( G ) then c ( x ) = c ( t ) ⇒ c ( y ) ≠ c ( z ) . The oriented chromatic number χ → ( G ) of an oriented graph G is defined as the smallest k such that G admits an oriented k-coloring. We prove in this paper that every Halin graph has oriented chromatic number at most 9, improving a previous bound proposed by Vignal.

A 3-approximation for the pathwidth of Halin graphs

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.

A 3-approximation for the pathwidth of Halin graphs

Abstract We prove that the pathwidth of Halin graphs can be approximated within a factor of 3

Maximum Genus of Strong Embeddings

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover. Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.

Constrained Steiner trees in Halin graphs

In this paper, we study the problem of computing a minimum cost Steiner tree subject to a weight constraint in a Halin graph where each edge has a nonnegative integer cost and a nonnegative integer weight. We prove the NP-hardness of this problem and present a fully polynomial time approximation scheme for this NP-hard problem.

Minimum cycle bases of Halin graphs

AbstractHalin graphs are planar 3‐connected graphs that consist of a tree and a cycle connecting the end vertices of the tree. It is shown that all Halin graphs that are not “necklaces” have a unique minimum cycle basis. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 150–155, 2003

The incidence coloring number of Halin graphs and outerplanar graphs

In this paper, we shall prove that the incidence coloring number for Halin graphs of maximum degree greater than or equal to 5 and outerplanar graphs of maximum degree greater than or equal to 4 is equal to the maximum degree of the graph plus 1.

On the edge-binding number of some plane graphs

LetG(V,E) be a simple graph, the edge-binding number b1 (G) ofG is defined as $$b_1 (G) = \min \left\{ {\left. {\frac{{|N(S)|}}{{|S|}}} \right|\emptyset \ne S \subset E,N(S) \ne E} \right\},$$ whereN(S) denotes the adjacent edges set ofS. In this paper, we obtained the edge-binding number of outer plane graphs, Halin graph and tree.

Contractible Subgraphs in 3-Connected Graphs

A subgraph H of a 3-connected finite graph G is called contractible if H is connected and G−V(H) is 2-connected. This work is concerned with a conjecture of McCuaig and Ota which states that for any given k there exists an f(k) such that any 3-connected graph on at least f(k) vertices possesses a contractible subgraph on k vertices. We prove this for k⩽4 and consider restrictions to maximal planar graphs, Halin graphs, line graphs of 6-edge-connected graphs, 5-connected graphs of bounded degree, and AT-free graphs.

On Steiner 2-edge connected polytopes

Abstract In this paper, we study the Steiner 2-edge connected subgraph polytope. We introduce a large class of valid inequalities for this polytope called generalized Steiner -P-partition inequalities, that generalize the so-called Steiner F -partition inequalities. We show that these inequalities together with the trivial and the Steiner cuts inequalities completely describe the polytope when the graph is a wheel. Using this, we discuss some applications to Halin graphs.

Persistency in the Traveling Salesman Problem on Halin graphs

Persistency in the Traveling Salesman Problem on Halin graphs

Hamiltonicity of Amalgams

An amalgam is obtained from two Halin graphs having skirting cycles of the same length. We are interested in hamiltonicity of amalgams constructed from two identical Halin graphs without any shift along the skirting cycle. We establish hamiltonicity of amalagams constructed from cubic Halin graphs. We give a sufficient condition for hamiltonicity of non-cubic amalgams and characterize infinite classes of non-Hamiltonian amalgams. We also characterize hamiltonicity of amalgams constructed by shifting the component Halin graphs by one and of general amalgams of higher degree.

A note on the total chromatic number of Halin graphs with maximum degree 4

In this paper, we prove that χT ( G) = 5 for any Halin graph G with Δ( G) = 4, where Δ( G) and χT ( G) denote the maximal degree and the total chromatic number of G, respectively.

A linear time algorithm for the bottleneck traveling salesman problem on a Halin graph

We provide a linear time algorithm to solve the bottleneck traveling salesman problem on a Halin graph. This improves the O( n log n) time bound obtained by the binary search version of the threshold algorithm.

The vertex-face total chromatic number of Halin graphs

In this paper, we shall first prove that for a Halin graph G, 4 ≤ χT(G) ≤ 6, where χT(G) is the vertex-face total chromatic number of G. Second, we shall establish a sufficient condition for a Halin graph to have a vertex-face total chromatic number of 6. Finally, we shall give a necessary and sufficient condition for a Halin graph to have a vertex-face total chromatic number of 4 and describe a way of assigning the four colors to such graphs. © 1997 John Wiley & Sons, Inc. Networks 30:167–170, 1997

Minimum-weight cycles in 3-separable graphs

This paper presents a polynomial-time algorithm for the minimum-weight-cycle problem on graphs that decompose via 3-separations into well-structured graphs. The problem is NP-hard in general. Graphs that decompose via 3-separations into well-structured graphs include Halin, outer-facial, delta-wye, wye-delta, flat, and twirl-wheel graphs. For each of these classes of graphs, given the decomposition, the algorithm runs in linear time.