The radius of -connected planar graphs with bounded faces (original) (raw)

The radius ofk-connected planar graphs with bounded faces

Discrete Mathematics, 2012

We prove that if G is a 3-connected plane graph of order p, maximum face length l and radius rad(G), then the bound rad(G) ≤ p 6 + 5l 6 + 2 3 holds. For constant l, our bound is shown to be asymptotically sharp and improves on a bound by Harant (1990) [6]. Furthermore we extend these results to 4-and 5-connected planar graphs.

Large planar graphs with given diameter and maximum degree

Discrete Applied Mathematics, 1995

We consider the problem of determining the maximum number of vertices in a planar graph with given maximum degree A and diameter k. This number has previously been exactly determined when k = 2. We show here that when k = 3, the number is roughly between 4.54 and 84. We also show that in general the number is @(A~izl) for any fixed value of k. -218X/95/$09.50 0 1995-Elsevier Science B.V. All rights reserved SSDI 0166-218X(94)00011-2 Corollary 5. If 1 D,, 1 > A -2, then at most three vertices of C,, have private neighbours in A,, with respect to C,, Proof. Suppose aI, . . . , ak, respectively, are private neighbours in A,, of vl, . . . , uk E C,, with respect to C,,. The cycle C, separates ai from Dxy, so in order for ai to be of distance at most three from each vertex of Dxy, Vi must be of distance two from each vertex of D,, By Lemma 4, k < 3. q

Steiner diameter of 3, 4 and 5-connected maximal planar graphs

Discrete Applied Mathematics, 2014

Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set con- Steiner distance of any n-subset of vertices of G. This is a generalisation of the ordinary diameter, which is the case n = 2. We give upper bounds on the Steiner n-diameter of maximum planar graphs in terms of order and connectivity. Moreover, we construct graphs to show that the bound is asymptotically sharp. Furthermore we extend this result to 4 and 5-connected maximal planar graphs.

Bounds on the number of cycles of length three in a planar graph

Israel Journal of Mathematics, 1982

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF 1,F 2, …,F r. Letf 1,f 2, …,f r denote the lengths of the cycles bounding the facesF 1,F 2, …,F r respectively. LetC 3(G) be the number of cycles of length three inG. We give bounds onC 3(G) in terms ofp,f 1,f 2, …,f r.

Domination in planar graphs with small diameter

Journal of Graph Theory, 2002

MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213-229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a unique planar graph of diameter two with domination number three, and all other planar graphs of diameter two have domination number at most two. We also prove that every planar graph of diameter three and of radius two has domination number at most six. We then show that every sufficiently large planar graph of diameter three has domination number at most seven. Analogous results for other surfaces are discussed.

A bound on the treewidth of planar even-hole-free graphs

Discrete Applied Mathematics, 2010

The class of planar graphs has unbounded treewidth, since the k × k grid, ∀k ∈ N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.

Even-hole-free planar graphs have bounded treewidth

Electronic Notes in Discrete Mathematics, 2008

The class of planar graphs has unbounded treewidth, since the k × k grid, ∀k ∈ N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an evenhole-free planar graph, then it does not contain a 9 × 9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 44.

Large induced forests in planar graphs with girth 4

Discrete Applied Mathematics, 2019

We give here some new lower bounds on the order of a largest induced forest in planar graphs with girth 4 and 5. In particular we prove that a triangle-free planar graph of order n admits an induced forest of order at least 6n+7 11 , improving the lower bound of Salavatipour [M. R. Salavatipour, Large induced forests in trianglefree planar graphs, Graphs and Combinatorics, 22:113-126, 2006]. We also prove that a planar graph of order n and girth at least 5 admits an induced forest of order at least 44n+50 69 .

On size, radius and minimum degree

Discrete Mathematics & Theoretical Computer Science, 2014

Graph Theory Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, radius and minimum degree. Our result is a strengthening of an old classical theorem of Vizing (1967) if minimum degree is prescribed.

On the Complexity of Covering Vertices by Faces in a Planar Graph

SIAM Journal on Computing, 1988

The pair (G, D) consisting of a planar graph G V, E) with n vertices together with a subset of d special vertices D V is called k-planar if there is an embedding of G in the plane so that at most k faces of G are required to cover all of the vertices in D. Checking 1-planarity can be done in linear-time since it reduces to a problem of checking planarity of a related graph. We present an algorithm which given a graph G and a value k either determines that G is not k-planar or generates an appropriate embedding and associated minimum cover in O(ckn) time, where c is a constant. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, d 0(n), and all facial cycles have bounded length. These results provide a polynomial-time recognition algorithm for special cases of Steiner tree problems in graphs which are solvable in polynomial time. Key words, complexity, planar graphs, Steiner trees AMS(MOS) subject classifications. 05, 68 1. Introduction. Recently, there has been a great deal of interest in solving the Steiner tree problem in graphs. This problem is NP-complete even for planar grid graphs [GJ1]. (See [GJ2] for an excellent introduction to the area of computational complexity.) So recent work has centered on efficiently-solvable special cases and heuristic methods; see [Wi] for a survey of work on this problem. Throughout this paper we deal with undirected graphs of the form G (V, E), where V is a set of n vertices and E is a set of edges connecting pairs of vertices. A graph is called planar if it can be embedded in the plane. A graph G V, E) together with d special vertices D V is called k-planar if there is a 131anar embedding of G so that at most k faces of G are required to cover all of the vertices in D. Clearly, a planar graph is the same as an n-planar graph. The planarity number of G is the minimum k such that G is k-planar. A recent paper by [EMV] presents an algorithm which solves the Steiner problem in an arbitrary graph; their algorithm runs in polynomial time for k-planar graphs, for any fixed k, with D being the vertices required to be in the Steiner tree. It is easy to see that checking 1-planarity of G V, E) with special vertices D V is equivalent to testing the planarity of the associated graph G*= (V*, E*), where V*= Vt.J {r} and E* E [_J {(r, v)" v D}, and so can be done in linear time [HT2]. They leave as an open question the complexity of testing k-planarity for fixed k->-2. In 2, we present an algorithm which checks to see if a given (G, D) pair is k-planar given a fixed embedding of G and if so, determines the planarity number of G in O(ckn) time, when c is a constant. This is used in 3 to generate an appropriate embedding of G and a cover of D by k or fewer faces, if possible, in O(ckn) time. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required grows exponentially in k is to be expected as we show in 4 that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, d O(n), and all facial cycles have bounded length.