Halin Graphs Research Articles

Dirac's Condition for Spanning Halin Subgraphs

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results(see, e.g., [J. A. Bondy, Handbook of Combinatorics, Vol. 1, MIT Press, Cambridge, MA, 1995, pp. 3--110]. A Halin graph is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least four vertices and with no vertex of degree 2, and a cycle induced by the set of the leaves of the spanning tree. Halin graphs possess rich hamiltonicity properties such as being hamiltonian, hamiltonian connected, and almost pancyclic. As a continuous “generalization” of Dirac's theorem, in this paper, we show that there exists a positive integer $n_0$ such that any graph $G$ with $n\ge n_0$ vertices and $\delta(G)\ge (n+1)/2$ contains a spanning and pancyclic Halin subgraph $H$. In addition, for every nonhamiltonian cycle $C$ in $H$, there is a cycle $C'$ longer than $C$ such that $C'$ contains all vertices from $C$ and at most two more vertices not from $C$.

Some further results on the eccentric distance sum

Let G = ( V ( G ) , E ( G ) ) be a simple connected graph. Then the eccentric distance sum of G , which is a novel graph invariant by offering a great potential for structure activity/property relationships, is defined as ξ d ( G ) = ∑ v ∈ V ε G ( v ) D G ( v ) , where ε G ( v ) is the eccentricity of the vertex v , and D G ( v ) is the sum of all distance from the vertex v . As a continuation to the parts of [22] (Li et al., J. Math. Anal. Appl. 43:1149–1162, 2015), and [17] (Hua et al., Discrete Appl. Math. 160:170–180, 2012), this paper answers one of some remaining problems in [22] is how to determine the bipartite graphs of even diameter with the minimum EDS, and gives a Nordhaus–Gaddum bound for eccentric distance sum, which is the generalization of the corresponding results in [17] . Moreover, some sharp lower bounds on the EDS of Halin graphs, and of triangle-free and quadrangle-free graphs are respectively presented.

Track Layouts, Layered Path Decompositions, and Leveled Planarity

We investigate two types of graph layouts, track layouts and layered path decompositions, and the relations between their associated parameters track-number and layered pathwidth. We use these two types of layouts to characterize leveled planar graphs, which are the graphs with planar leveled drawings with no dummy vertices. It follows from the known NP-completeness of leveled planarity that track-number and layered pathwidth are also NP-complete, even for the smallest constant parameter values that make these parameters nontrivial. We prove that the graphs with bounded layered pathwidth include outerplanar graphs, Halin graphs, and squaregraphs, but that (despite having bounded track-number) series–parallel graphs do not have bounded layered pathwidth. Finally, we investigate the parameterized complexity of these layouts, showing that past methods used for book layouts do not work to parameterize the problem by treewidth or almost-tree number but that the problem is (non-uniformly) fixed-parameter tractable for tree-depth.

Realizability of graphs as triangle cover contact graphs

Let S={p1,p2,…,pn} be a set of pairwise disjoint geometric objects of some type and let C={c1,c2,…,cn} be a set of closed objects of some type with the property that each element in C covers exactly one element in S and any two elements in C can intersect only on their boundaries. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) consists of a set of vertices and a set of edges where each of the vertex corresponds to each of the covers and each edge corresponds to a connection between two covers if and only if they touch at their boundaries. A triangle cover contact graph (TCCG) is a cover contact graph whose cover elements are triangles. In this paper, we show that every Halin graph has a realization as a TCCG on a given set of collinear seeds. We introduce a new class of graphs which we call super-Halin graphs. We also show that the classes super-Halin graphs, cubic planar Hamiltonian graphs and a×b grid graphs have realizations as TCCGs on collinear seeds. We also show that some non-planar graphs have planar realizations as TCCGs. Every complete graph and clique-3 graph have realizations as TCCGs on any given set of seeds. Note that only trees and cycles are known to be realizable as CCGs and outerplanar graphs are known to be realizable as TCCGs.

Listing all spanning trees in Halin graphs — sequential and Parallel view

For a connected labeled graph [Formula: see text], a spanning tree [Formula: see text] is a connected and acyclic subgraph that spans all vertices of [Formula: see text]. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of [Formula: see text]. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of [Formula: see text] processors for parallel algorithmics, where [Formula: see text] and [Formula: see text] are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is [Formula: see text].

Upper bounds for the strong chromatic index of Halin graphs

Upper bounds for the strong chromatic index of Halin graphs

A linear time algorithm for the [formula omitted]-neighbour Travelling Salesman Problem on a Halin graph and extensions

The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamiltonian cycle in an edge-weighted graph, where costs are defined for all pairs of edges contained in the Hamiltonian cycle. The problem is shown to be strongly NP-hard on a Halin graph. We also consider a variation of the QTSP, called the k-neighbour TSP (TSP(k)). Two edges e and f, e≠f, are k-neighbours on a tour τ if and only if a shortest path (with respect to the number of edges) between e and f along τ and containing both e and f, has exactly k edges, for k≥2. In (TSP(k)), a fixed nonzero cost is considered for a pair of distinct edges in the cost of a tour τ only when the edges are p-neighbours on τ for 2≤p≤k. We give a linear time algorithm to solve TSP(k) on a Halin graph for k=3, extending existing algorithms for the cases k=1,2. Our algorithm can be extended further to solve TSP(k) in polynomial time on a Halin graph with n nodes when k=O(logn). The possibility of extending our results to some fully reducible class of graphs is also discussed. TSP(k) can be used to model the Permuted Variable Length Markov Model in bioinformatics as well as an optimal routing problem for unmanned aerial vehicles (UAVs).

Forbidden Pairs and the Existence of a Spanning Halin Subgraph

A Halin graph is constructed from a plane embedding of a tree with no vertices of degree 2 by adding a cycle through its leaves in the natural order determined by the embedding. Halin graphs satisfy interesting properties. However, to our knowledge, there are no results giving a positive answer for “spanning Halin subgraph problem” (i.e., which graph has a Halin graph as a spanning subgraph) except for a conjecture by Lovasz and Plummer which states that every 4-connected plane triangulation contains a spanning Halin subgraph. In this paper, we investigate the characterization of forbidden pairs guaranteeing the existence of a spanning Halin subgraph. In particular, we show that the set of such pairs is a very small class. Also, we show that $$\{ K_{1,3}, P_5 \}$$ belongs to the set, but neither $$\{ K_{1,4}, P_5 \}$$ nor $$\{ K_{1,3}, P_6 \}$$ belongs to the set.

On the Laplacian spectral radii of Halin graphs

Let T be a tree with at least four vertices, none of which has degree 2, embedded in the plane. A Halin graph is a plane graph constructed by connecting the leaves of T into a cycle. Thus the cycle C forms the outer face of the Halin graph, with the tree inside it. Let G be a Halin graph with order n. Denote by mu(G) the Laplacian spectral radius of G. This paper determines all the Halin graphs with mu(G)geq n-4. Moreover, we obtain the graphs with the first three largest Laplacian spectral radius among all the Halin graphs on n vertices.

Permanent Index of Matrices Associated with Graphs

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $\phi_f(u) \ne \phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. This approach leads to conjectures on the permanent indices of matrices $A_G$ and $B_G$ associated to a graph $G$. In this paper, we establish a method that reduces the study of permanent of matrices associated to a graph $G$ to the study of permanent of matrices associated to induced subgraphs of $G$. Using this reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable.

The decycling number and vertex coloring of Halin graphs

The decycling number and vertex coloring of Halin graphs

Plane Triangulations Without a Spanning Halin Subgraph II

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 remaining Hamiltonian after any single vertex deletion. In 1975, Lovasz and Plummer conjectured that every 4-connected plane triangulation contains a spanning Halin subgraph. We recently gave a negative answer to this conjecture. In this paper, we construct an infinite class of 5-connected plane triangulations without a spanning Halin subgraph. Our smallest example contains 512 vertices.

Competition Numbers of a Kind of Pseudo-Halin Graphs

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 chara-cterizing a graph by its competition number has been one of important research problems in the study of competition graphs. A 2-connected planar graph G with minimum degree at least 3 is a pseudo-Halin graph if deleting the edges on the boundary of a single face f0 yields a tree. It is a Halin graph if the vertices of f0 all have degree 3 in G. In this paper, we compute the competition numbers of a kind of pseudo-Halin graphs.

VPG and EPG bend-numbers of Halin graphs

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-numberbv(G) of a graph G is the minimum k such that G has a Bk-VPG representation. Similarly, the EPG bend-numberbe(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.

Extremal Halin graphs with respect to the signless Laplacian spectra

A Halin graph G is a plane graph constructed as follows: Let T be a tree on at least 4 vertices. All vertices of T are either of degree 1, called leaves, or of degree at least 3. Let C be a cycle connecting the leaves of T in such a way that C forms the boundary of the unbounded face. Denote the set of all n-vertex Halin graphs by Gn. In this article, sharp upper and lower bounds on the signless Laplacian indices of graphs among Gn are determined and the extremal graphs are identified, respectively. As well graphs in Gn having the second and third largest signless Laplacian indices are determined, respectively.

Simple Recognition of Halin Graphs and Their Generalizations

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, finding a Hamiltonian cycle, or constructing a planar embedding. These reduction rules can also be used to recognize a broader class of polyhedral graphs. These graphs, which we call the D3-reducible graphs, are the dual graphs of the polyhedra formed by gluing pyramids together on their triangular faces; their treewidth is bounded, and they necessarily have Lombardi drawings.

Induced Cycle Path Number of Graphs

Let G=(V, E) be a simple connected graph. An induced cycle path partition of G is a partition of V (G) into subsets such that the subsets induce cycles and paths. The minimum order taken over all induced cycle path partitions is called the induced cycle path number of G and is denoted by ρcp(G). In this paper, we dertermine this parameter for some standard graphs and investigate the same for unicyclic graphs and Halin graphs. We also obtain an upper bound for arbitrary graphs and characterize the extremal graphs. The graphs which have unique induced cycle path partition are also discussed

Total weight choosability of Mycielski graphs

A total weighting of a graph G is a mapping $$\phi $$ź that assigns a weight to each vertex and each edge of G. The vertex-sum of $$v \in V(G)$$vźV(G) with respect to $$\phi $$ź is $$S_{\phi }(v)=\sum _{e\in E(v)}\phi (e)+\phi (v)$$Sź(v)=źeźE(v)ź(e)+ź(v). A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph $$G=(V,E)$$G=(V,E) is called $$(k,k')$$(k,kź)-choosable if the following is true: If each vertex x is assigned a set L(x) of k real numbers, and each edge e is assigned a set L(e) of $$k'$$kź real numbers, then there is a proper total weighting $$\phi $$ź with $$\phi (y)\in L(y)$$ź(y)źL(y) for any $$y \in V \cup E$$yźVźE. In this paper, we prove that for any graph $$G\ne K_1$$GźK1, the Mycielski graph of G is (1,4)-choosable. Moreover, we give some sufficient conditions for the Mycielski graph of G to be (1,3)-choosable. In particular, our result implies that if G is a complete bipartite graph, a complete graph, a tree, a subcubic graph, a fan, a wheel, a Halin graph, or a grid, then the Mycielski graph of G is (1,3)-choosable.

A Note on Total and List Edge-Colouring of Graphs of Tree-Width 3

It is shown that Halin graphs are $$\varDelta $$Δ-edge-choosable and that graphs of tree-width 3 are $$(\varDelta +1)$$(Δ+1)-edge-choosable and $$(\varDelta +2)$$(Δ+2)-total-colourable.

Hamiltonian decomposition of prisms over cubic graphs

Special issue PRIMA 2013 The prisms over cubic graphs are 4-regular graphs. The prisms over 3-connected cubic graphs are Hamiltonian. In 1986 Brian Alspach and Moshe Rosenfeld conjectured that these prisms are Hamiltonian decomposable. In this paper we present a short survey of the status of this conjecture, various constructions proving that certain families of prisms over 3-connected cubic graphs are Hamiltonian decomposable. Among others, we prove that the prisms over cubic Halin graphs, cubic generalized Halin graphs of order 4k + 2 and other infinite sequences of cubic graphs are Hamiltonian decomposable.