Subdivisions of graphs: A generalization of paths and cycles

One of the basic results in graph theory is Dirac's theorem, that every graph of order n?3 and minimum degree ?n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree ?n/2 contains a cycle C then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every connected component of H is non-trivial and contains at most one cycle. The degree bound can be improved to (n-t)/2 if H has t components that are trees.We attempt a similar generalization of the Corrádi-Hajnal theorem that every graph of order ?3k and minimum degree ?2k contains k disjoint cycles. Again, this may be restated as: every graph of order ?3k and minimum degree ?2k contains a subdivision of kK₃. We show that if H is any graph of order n with k components, each of which is a cycle or a non-trivial tree, then every graph of order ?n and minimum degree ?n-k contains a subdivision of H.     

Improved heuristics for the minimum weight triangulation problem

Given a planar point setS, a triangulation ofS is a maximal set of non-intersecting line segments connecting the points. The minimum weight triangulation problem is to find a triangulation ofS such that the sum of the lengths of the line segments in it is the smallest. No polynomial time algorithm is known to produce the optimal or even a constant approximation of the optimal solution, and it is also unknown whether the problem is NP-hard. In this paper, we propose two improved heuristics, which triangulate a set ofn points in a plane inO(n ³) time and never do worse than the minimum spanning tree triangulation algorithm given by Lingas and the greedy spanning tree triangulation algorithm given by Heath and Pemmaraju. These two algorithms both produce an optimal triangulation if the points are the vertices of a convex polygon, and also do the same in some special cases.     

Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

On the number of Hamiltonian cycles in triangulations

It is proved that if a planar triangulation different from K₃ and K₄ contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [2], this yields that, for n ? 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3-connected triangulated graphs.     

Quasi-Greedy Triangulations Approximating the Minimum Weight Triangulation

This article settles the following two longstanding open problems:

• What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation?

• Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?

The answer to the first question is that the known lower bound is tight. The second question is answered in the affirmative by using a slight modification of anO(n log n) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds.     

A new asymmetric inclusion region for minimum weight triangulation

As a global optimization problem, planar minimum weight triangulation problem has attracted extensive research attention. In this paper, a new asymmetric graph called one-sided β-skeleton is introduced. We show that the one-sided circle-disconnected (?2b){(\sqrt{2}\beta)} -skeleton is a subgraph of a minimum weight triangulation. An algorithm for identifying subgraph of minimum weight triangulation using the one-sided (?2b){(\sqrt{2}\beta)} -skeleton is proposed and it runs in O(n^4/3+e+min{klogn, n²logn}){O(n^{4/3+\epsilon}+\min\{\kappa \log n, n^2\log n\})} time, where κ is the number of intersected segmented between the complete graph and the greedy triangulation of the point set.     

Polynomial area bounds for MST embeddings of trees

In their seminal paper on geometric minimum spanning trees, Monma and Suri (1992) [31] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane. The embeddings provided by their algorithm require area O(n²²)×O(n²²) and the authors conjectured that an improvement below cn×cn is not possible, for some constant c>0. In this paper, we show how to construct MST embeddings of arbitrary trees of maximum degree 3 and 4 within polynomial area.     

Some results on odd factors of graphs

A {1, 3, …,2n ? 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, …, 2n ? 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, …, 2n ? 1}-factor.     

Enumerating edge-constrained triangulations and edge-constrained non-crossing geometric spanning trees

In this paper, we present algorithms for enumerating, without repetitions, all triangulations and non-crossing geometric spanning trees on a given set of n points in the plane under edge inclusion constraint (i.e., some edges are required to be included in the graph). We will first extend the lexicographically ordered triangulations introduced by Bespamyatnikh to the edge-constrained case, and then we prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the edge-constrained lexicographically largest triangulation. More specifically, we prove that all edge-constrained triangulations can be transformed to the lexicographically largest triangulation among them by O(n²) greedy flips, i.e., by greedily increasing the lexicographical ordering of the edge list, and a similar result also holds for a set of edge-constrained non-crossing spanning trees. Our enumeration algorithms generate each output triangulation and non-crossing spanning tree in O(loglogn) and O(n²) time, respectively, based on the reverse search technique.     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011     

Diagonal Flips in Labelled Planar Triangulations

 A classical result of Wagner states that any two (unlabelled) planar triangulations with the same number of vertices can be transformed into each other by a finite sequence of diagonal flips. Recently Komuro gives a linear bound on the maximum number of diagonal flips needed for such a transformation. In this paper we show that any two labelled triangulations can be transformed into each other using at most O(nlogn) diagonal flips. We will also show that any planar triangulation with n>4 vertices has at least n− 2 flippable edges. Finally, we prove that if the minimum degree of a triangulation is at least 4, then it contains at least 2n + 3 flippable edges. These bounds can be achieved by an infinite class of triangulations. Received: June 3, 1998 Final version received: January 26, 2001     

Compact floor-planning via orderly spanning trees

Floor-planning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)-time algorithm to construct a floor-plan for any n-node plane triangulation. In comparison with previous floor-planning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floor-plans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floor-plan area in a rectangle of size (n−1)×(2n+1)/3. Lower bounds on the worst-case area for floor-planning any plane triangulation are also provided in the paper.     

Graphs with homeomorphically irreducible spanning trees

It is an NP-complete problem to decide whether a graph contains a spanning tree with no vertex of degree 2. We show that these homeomorphically irreducible spanning trees are contained in all graphs with minimum degree at least c√n and in triangulations of the plane. They are nearly present in all graphs of diameter 2. They do not necessarily occur in r-regular or r-connected graphs.     

Claw‐free graphs and 2‐factors that separate independent vertices

In this article, we prove that a line graph with minimum degree δ≥7 has a spanning subgraph in which every component is a clique of order at least three. This implies that if G is a line graph with δ≥7, then for any independent set S there is a 2‐factor of G such that each cycle contains at most one vertex of S. This supports the conjecture that δ≥5 is sufficient to imply the existence of such a 2‐factor in the larger class of claw‐free graphs. It is also shown that if G is a claw‐free graph of order n and independence number α with δ≥2n/α?2 and n≥3α³/2, then for any maximum independent set S, G has a 2‐factor with α cycles such that each cycle contains one vertex of S. This is in support of a conjecture that δ≥n/α≥5 is sufficient to imply the existence of a 2‐factor with α cycles, each containing one vertex of a maximum independent set. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 251–263, 2012     

Spanning 2‐trails from degree sum conditions

Suppose G is a simple connected n‐vertex graph. Let σ₃(G) denote the minimum degree sum of three independent vertices in G (which is ∞ if G has no set of three independent vertices). A 2‐trail is a trail that uses every vertex at most twice. Spanning 2‐trails generalize hamilton paths and cycles. We prove three main results. First, if σ₃G)≥ n ‐ 1, then G has a spanning 2‐trail, unless G ? K_1,3. Second, if σ₃(G) ≥ n, then G has either a hamilton path or a closed spanning 2‐trail. Third, if G is 2‐edge‐connected and σ₃(G) ≥ n, then G has a closed spanning 2‐trail, unless G ? K_2,3 or K (the 6‐vertex graph obtained from K_2,3 by subdividing one edge). All three results are sharp. These results are related to the study of connected and 2‐edge‐connected factors, spanning k‐walks, even factors, and supereulerian graphs. In particular, a closed spanning 2‐trail may be regarded as a connected (and 2‐edge‐connected) even [2,4]‐factor. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 298–319, 2004     

On Minimum Edge Ranking Spanning Trees

In this paper, we introduce the problem of computing a minimum edge ranking spanning tree (MERST); i.e., find a spanning tree of a given graph G whose edge ranking is minimum. Although the minimum edge ranking of a given tree can be computed in polynomial time, we show that problem MERST is NP-hard. Furthermore, we present an approximation algorithm for MERST, which realizes its worst case performance ratio where n is the number of vertices in G and Δ* is the maximum degree of a spanning tree whose maximum degree is minimum. Although the approximation algorithm is a combination of two existing algorithms for the restricted spanning tree problem and for the minimum edge ranking problem of trees, the analysis is based on novel properties of the edge ranking of trees.     

Computing a Minimum Weight Triangulation of a Sparse Point Set*

Investigating the minimum weight triangulation of a point set with constraint is an important approach for seeking the ultimate solution of the minimum weight triangulation problem. In this paper, we consider the minimum weight triangulation of a sparse point set, and present an O(n ⁴) algorithm to compute a triangulation of such a set. The property of sparse point set can be converted into a new sufficient condition for finding subgraphs of the minimum weight triangulation. A special point set is exhibited to show that our new subgraph of minimum weight triangulation cannot be found by any currently known methods.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

Spanning k‐arc‐strong subdigraphs with few arcs in k‐arc‐strong tournaments

Given a k‐arc‐strong tournament T, we estimate the minimum number of arcs possible in a k‐arc‐strong spanning subdigraph of T. We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k‐arc‐strong spanning subdigraph of T contains at least arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has arcs. This is best possible since it can be shown that every k‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than arcs. As our main result we prove that every k‐arc‐strong tournament contains a spanning k‐arc‐strong subdigraph with no more than arcs. We conjecture that for every k‐arc‐strong tournament T, the minimum number of arcs in a k‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k. We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004