Minor-closed Family Of Graphs Research Articles

Graph coloring with no large monochromatic components

Abstract For a graph G and an integer t we let m c c t ( G ) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc 2 ( G ) = O ( n 2 / 3 ) for any n-vertex graph G ∈ F . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F , and every fixed t we show that mcc t ( G ) = O ( n 2 / ( t + 1 ) ) . On the other hand, we have examples of graphs G with no K t + 3 minor and with mcc t ( G ) = Ω ( n ( 2 / 2 t − 1 ) ) . It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved mcc 2 ( G ) ⩽ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc 2 ( G ) = Ω ( n ) , and more sharply, for every e > 0 there exists c e > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + e for all subgraphs, and with mcc 2 ( G ) ⩾ c e n . For 6-regular graphs it is known only that the maximum order of magnitude of mcc 2 is between n and n. We also offer a Ramsey-theoretic perspective of the quantity m c c t ( G ) .

Oracles for bounded-length shortest paths in planar graphs

We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = ( V , E ), one can build in O(|V|) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed.Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service.Our results can be easily generalized to other wide classes of graphs---for instance we can take any minor-closed family of graphs.

Layout of Graphs with Bounded Tree-Width

A \emph{queue layout} of a graph consists of a total order of the vertices, and a partition of the edges into \emph{queues}, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line grid) drawing} of a graph represents the vertices by points in $\mathbb{Z}^3$ and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph $G$ is closely related to the queue-number of $G$. In particular, if $G$ is an $n$-vertex member of a proper minor-closed family of graphs (such as a planar graph), then $G$ has a $O(1)\times O(1)\times O(n)$ drawing if and only if $G$ has O(1) queue-number. (2) It is proved that queue-number is bounded by tree-width, thus resolving an open problem due to Ganley and Heath (2001), and disproving a conjecture of Pemmaraju (1992). This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with O(n) volume. This is the most general family of graphs known to admit three-dimensional drawings with O(n) volume. The proofs depend upon our results regarding \emph{track layouts} and \emph{tree-partitions} of graphs, which may be of independent interest.

Diameter and Treewidth in Minor-Closed Graph Families, Revisited

Eppstein [5] characterized the minor-closed graph families for which the treewidth is bounded by a function of the diameter, which includes, e.g., planar graphs. This characterization has been used as the basis for several (approximation) algorithms on such graphs (e.g., [2] and [5]–[8]). The proof of Eppstein is complicated. In this short paper we obtain the same characterization with a simple proof. In addition, the relation between treewidth and diameter is slightly better and explicit.

Bidimensional Parameters and Local Treewidth

For several graph-theoretic parameters such as vertex cover and dominating set, it is known that their sizes are bounded by k, then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded genus. In this paper we examine whether similar bounds can be obtained for larger minor-closed graph classes and for general families of graph parameters, including all those for which such behavior has been reported so far. Given a graph parameter P, we say that a graph family $\mathcal{F}$ has the parameter-treewidth property for P there is an increasing function t such that every graph $G\in\mathcal{F}$ has treewidth at most t(P(G)). We prove as our main result that, for a large family of graph parameters called contraction-bidimensional, a minor-closed graph family $\mathcal{F}$ has the parameter-treewidth property $\mathcal{F}$ has bounded local treewidth. We also show and only if for some graph parameters, and thus, this result is in some sense tight. In addition we show that, for a slightly smaller family of graph parameters called minor-bidimensional, all minor-closed graph families $\mathcal{F}$, excluding some fixed graphs, have the parameter-treewidth property. The contraction-bidimensional parameters include many domination and covering graph parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, and q-dominating set (for fixed q). We use our theorems to develop new fixed-parameter algorithms in these contexts.

Forbidden minors to graphs with small feedback sets

Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the graph minor theorem. In this paper we characterize several families of graphs with small feedback sets, namely k 1-F EEDBACK V ERTEX S ET, k 2-F eedback E DGE S ET and ( k 1, k 2)-F EEDBACK V ERTEX/E DGE S ET, for small integer parameters k 1 and k 2. Our constructive methods can compute obstruction sets for any minor-closed family of graphs, provided the pathwidth (or treewidth) of the largest obstruction is known.

Bounding the Size of Planar Intertwines

The proof of Wagner's conjecture by Robertson and Seymour gives a finite description of any family of graphs which is closed under the minor ordering. This description is a finite set of minimal graphs not in the family; these graphs are called the obstructions of the family. Since the intersection and union of two minor closed graph families is again a minor closed graph family, an interesting question regards computing the obstructions of the new family given the obstructions for the original two families. It is easy to compute the obstructions of the intersection, but nontrivial to compute those of the union. In this paper, we show that if the original families are planar then the planar obstructions of the union are no larger than nO(n2), where n is the size of the largest obstruction of the original families.