2-connected Outerplanar Graph Research Articles

Anti-van der Waerden Numbers of Some 2-Connected Outerplanar Graphs

Given a graph [Formula: see text] and a positive integer [Formula: see text], the anti-van der Waerden number [Formula: see text] is defined as the minimum positive integer [Formula: see text] such that every exact [Formula: see text]-coloring of the vertices of [Formula: see text] admits a rainbow [Formula: see text]-AP. In this paper, we determine the exact value of [Formula: see text] if [Formula: see text] is a [Formula: see text]-connected outerplanar graph with diameters [Formula: see text].

Uniquely colorable graphs up to automorphisms

We extend the concept of uniquely colorable graphs and say that a graph G is χ-iso-unique if for every two proper colorings c:V(G)→{1,…,χ(G)} and d:V(G)→{1,…,χ(G)} there exists an automorphism φ of G such that for any i∈{1,…,χ(G)} there exists j∈{1,…,χ(G)} so that φ maps c−1(i) onto d−1(j). First, we prove some general properties about degrees of vertices of χ-iso-unique graphs G, and then focus on non-bipartite outerplanar χ-iso-unique graphs. As proved by Chartrand and Geller back in 1969, uniquely 3-colorable outerplanar graphs are precisely the 2-connected outerplanar graphs in which all induced cycles are triangles. We prove that a χ-iso-unique outerplanar graph G with χ(G)=3 contains at most one inner face, which is not a triangle, and it is isomorphic to C5 if it exists. Besides maximal outerplanar graphs, there is just one additional family of 2-connected outerplanar graphs that are χ-iso-unique, and they contain one facial 5-cycle. In contrast to the situation in uniquely colorable graphs, we find two families of outerplanar χ-iso-unique graphs having cut-vertices, and finally, we characterize all outerplanar χ-iso-unique graphs. We also show that χ-iso-unique outerplanar graphs can be recognized efficiently.

Independent domination in outerplanar graphs

In this paper we consider the maximum value of the independent domination number of outerplanar graphs. If one considers all outerplanar graphs or restricts to all maximal outerplanar graphs, this maximum is easy to find and known. So we focus on some subfamilies. Our main contribution is that, if one restricts to 2-connected outerplanar graphs of order n , then the maximum is ( 2 n + 1 ) / 5 , and there exists an infinite family of graphs that achieve this bound whenever 2 n + 1 is a multiple of 5.

Crumby colorings — Red-blue vertex partition of subcubic graphs regarding a conjecture of Thomassen

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most 1 (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring crumby. Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky [2] constructed an infinite family refuting the above conjecture. Their prototype counterexample is 2-connected, planar, but contains a K4-minor and also a 5-cycle. This leaves the above conjecture open for some important graph classes: outerplanar graphs, K4-minor-free graphs, bipartite graphs. In this regard, we prove that 2-connected outerplanar graphs, subdivisions of K4 and 1-subdivisions of cubic graphs admit crumby colorings. A subdivision of G is genuine if every edge is subdivided at least once. We show that every genuine subdivision of any subcubic multigraph admits a crumby coloring. We slightly generalize some of these results and formulate a few conjectures.

The maximum average connectivity among all orientations of a graph

For distinct vertices u and v in a graph G, the connectivity between u and v, denoted \(\kappa _G(u,v)\), is the maximum number of internally disjoint u–v paths in G. The average connectivity of G, denoted \({\overline{\kappa }}(G),\) is the average of \(\kappa _G(u,v)\) taken over all unordered pairs of distinct vertices u, v of G. Analogously, for a directed graph D, the connectivity from u to v, denoted \(\kappa _D(u,v)\), is the maximum number of internally disjoint directed u–v paths in D. The average connectivity of D, denoted \({\overline{\kappa }}(D)\), is the average of \(\kappa _D(u,v)\) taken over all ordered pairs of distinct vertices u, v of D. An orientation of a graph G is a directed graph obtained by assigning a direction to every edge of G. For a graph G, let \({\overline{\kappa }}_{\max }(G)\) denote the maximum average connectivity among all orientations of G. In this paper we obtain bounds for \({\overline{\kappa }}_{\max }(G)\) and for the ratio \({\overline{\kappa }}_{\max }(G)/{\overline{\kappa }}(G)\) for all graphs G of a given order and in a given class of graphs. Whenever possible, we demonstrate sharpness of these bounds. This problem had previously been studied for trees. We focus on the classes of cubic 3-connected graphs, minimally 2-connected graphs, 2-trees, and maximal outerplanar graphs.

Packing [formula omitted]-coloring of subcubic outerplanar graphs

For 1≤s1≤s2≤…≤sk and a graph G, a packing(s1,s2,…,sk)-coloring of G is a partition of V(G) into sets V1,V2,…,Vk such that, for each 1≤i≤k the distance between any two distinct x,y∈Vi is at least si+1. The packing chromatic number, χp(G), of a graph G is the smallest k such that G has a packing (1,2,…,k)-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs G with arbitrarily large χp(G). Recently, there was a series of papers on packing (s1,s2,…,sk)-colorings of subcubic graphs in various classes. We show that every 2-connected subcubic outerplanar graph has a packing (1,1,2)-coloring and every subcubic outerplanar graph is packing (1,1,2,4)-colorable. Our results are sharp in the sense that there are 2-connected subcubic outerplanar graphs that are not packing (1,1,3)-colorable and there are subcubic outerplanar graphs that are not packing (1,1,2,5)-colorable. We also show subcubic outerplanar graphs that are not packing (1,2,2,4)-colorable and not packing (1,1,3,4)-colorable.

A note on a conjecture of star chromatic index for outerplanar graphs

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge coloring by $k$ colors. In \cite{LB}, L. Bezegov$\acute{a}$ et al. conjectured that $\chi_{st}^{'} (G )\leq \lfloor\frac{3\Delta}{2}\rfloor+1$ when $G$ is an outerplanar graph with maximum degree $\Delta \geq 3.$ In this paper we obtained that $\chi_{st}^{'}(G) \leq \Delta+6$ when $G$ is an 2-connected outerplanar graph with diameter 2 or 3. If $G$ is an 2-connected outerplanar graph with maximum degree 5, then $\chi_{st}^{'}(G) \leq 9.$

Resonance graphs of catacondensed even ring systems

A catacondensed even ring system (shortly CERS) is a simple bipartite 2-connected outerplanar graph with all vertices of degree 2 or 3. In this paper, we investigate the resonance graphs (also called Z-transformation graphs) of CERS and firstly show that two even ring chains are evenly homeomorphic iff their resonance graphs are isomorphic. As the main result, we characterize CERS whose resonance graphs are daisy cubes. In this way, we greatly generalize the result known for kinky benzenoid graphs. Finally, some open problems are also presented.

Minimum size tree-decompositions

We study in this paper the problem of computing a tree-decomposition of a graph with width at most k and minimum number of bags. More precisely, we focus on the following problem: given a fixed k≥1, what is the complexity of computing a tree-decomposition of width at most k with minimum number of bags in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k≥4 and polynomial for k≤2; for k=3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs.

The Strong Arnold Property for 4-connected flat graphs

We show that if G=(V,E) is a 4-connected flat graph, then any real symmetric V×V matrix M with exactly one negative eigenvalue and satisfying, for any two distinct vertices i and j, Mij<0 if i and j are adjacent, and Mij=0 if i and j are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric V×V matrix X with MX=0 and Xij=0 whenever i and j are equal or adjacent. (A graph G is flat if it can be embedded injectively in 3-dimensional Euclidean space such that the image of any circuit is the boundary of some disk disjoint from the image of the remainder of the graph.)This applies to the Colin de Verdière graph parameter, and extends similar results for 2-connected outerplanar graphs and 3-connected planar graphs.

Minimum Size Tree-decompositions

Tree-decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. We prove that the problem is NP-complete for any fixed k≥4 and polynomial for k≤2; for k=3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs.

Gallai's conjecture for outerplanar graphs

Let G be a connected simple graph on n vertices. The Gallai's conjecture asserts that the edges of G can be decomposed into paths. In this paper, we show that the conjecture is true if G is a maximal outerplanar graph or G ia a 2-connected outerplanar graph.

THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

Cycle Bases Structure of out Planar Graphs on the Projective Plane in Mechanics Engineering

In this paper we investigate the cycle base structure of 2-connected graphs on the projective plane and show the minimum cycle bases of 2-connected outer planar graph G in the case of ew (G) 5. Then give a proof about the one-one property between the minimum cycle bases and the shortest no contractible cycles.

Bounded Tolerance Representations for Maximal Outerplanar Graphs

A graph G = (V, E) is a tolerance graph if there is a set I = {I(subscript υ)|υ ∈ V} of closed real interval and a set т = { т(subscript υ)|υ ∈ V} of positive real numbers such that (x, y) ∈ E ↔ |I(subscript x)∩I(subscript y)| ≥ min{ т(subscript x), т(subscript y)}. We show that if G is a 2-connected maximal outer-planar graph with more than two vertices of degree 2, then G has S3 as an induced subgraph. We provide a characterization of the class of 2-connected maximal outer-planar graphs that are bounded tolerance graphs.

The Maximum Degree of Series-Parallel Graphs

We prove that the maximum degree Δnof a random series-parallel graph withnvertices satisfies Δn/logn→cin probability, andΔn~clognfor a computable constantc&gt; 0. The same kind of result holds for 2-connected series-parallel graphs, for outerplanar graphs, and for 2-connected outerplanar graphs.

Equitable coloring planar graphs with large girth

A proper vertex coloring of a graph G is equitable if the size of color classes differ by at most one. The equitable chromatic threshold of G, denoted by χ Eq * ( G ) , is the smallest integer m such that G is equitably n-colorable for all n ⩾ m . We prove that χ Eq * ( G ) = χ ( G ) if G is a non-bipartite planar graph with girth ⩾ 26 and δ ( G ) ⩾ 2 or G is a 2-connected outerplanar graph with girth ⩾ 4 .

Power domination in planar graphs with small diameter

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In this paper, it was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one. Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling, et al. Also, an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given.

A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs

An edge ranking of a graph G is a labeling r of its edges with positive integers such that every path between two different edges e u , e v with the same rank r ( e u ) = r ( e v ) contains an intermediate edge e w with rank r ( e w ) > r ( e u ) . An edge ranking of G is minimum if the largest rank k assigned is the smallest among all rankings of G. The edge ranking problem is to find a minimum edge ranking of given graph G. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of trees. In this paper, we first show, on a general graph G, a relation between a minimum edge ranking of G and its minimal cuts, which ensures that we can obtain a polynomial time algorithm for obtaining minimum edge ranking of a given graph G if minimal cuts for each subgraph of G can be found in polynomial time and the number of those is polynomial. Based on this relation, we develop a polynomial time algorithm for finding a minimum edge ranking on a 2-connected outerplanar graph.

Light graphs in families of outerplanar graphs

We prove that every 2-connected outerplanar graph of order at least k ( k ⩾ 3 ) contains a path on k vertices with all vertices of degree at most k + 3 and a path on k vertices with degree sum at most 4 k - 2 . Further, every 2-connected outerplanar graph without adjacent vertices with degree sum ⩽ 5 contains a triangle C 3 with vertices of degrees 2, 4 and b, 4 ⩽ b ⩽ 6 , and a 3-star K 1 , 3 with central vertex of degree 4 and remaining vertices of degrees 2, 2 and b, 4 ⩽ b ⩽ 6 . Moreover, all bounds are best possible.