Graph minors
- Connectivity, graph minors, and subgraph multiplicity.
D. Eppstein.
Tech. Rep. 92-06, ICS, UCI, 1992.
J. Graph Th. 17: 409–416, 1993.It was known that planar graphs have \(O(n)\) subgraphs isomorphic to \(K_3\) or \(K_4\). That is, \(K_3\) and \(K_4\) have linear subgraph multiplicity. This paper shows that the graphs with linear subgraph multiplicity in the planar graphs are exactly the 3-connected planar graphs. Also, the graphs with linear subgraph multiplicity in the outerplanar graphs are exactly the 2-connected outerplanar graphs.
More generally, let \(\mathcal{F}\) be a minor-closed family, and let \(x\) be the smallest number such that some complete bipartite graph \(K_{x,y}\) is a forbidden minor for \(\mathcal{F}\). Then the \(x\)-connected graphs have linear subgraph multiplicity for \(\mathcal{F}\), and there exists an \((x-1)\)-connected graph (namely \(K_{x-1,x-1}\) that does not have linear subgraph multiplicity. When \(x\le 3\) or when \(x=4\) and the minimal forbidden minors for \(\mathcal{F}\) are triangle-free, then the graphs with linear subgraph multiplicity for \(\mathcal{F}\) are exactly the \(x\)-connected graphs.
Please refer only to the journal version, and not the earlier technical report: the technical report had a bug that was repaired in the journal version.
- Subgraph isomorphism in planar graphs and related problems.
D. Eppstein.
Tech. Rep. 94-25, ICS, UCI, 1994.
6th ACM-SIAM Symp. Discrete Algorithms, San Francisco, 1995, pp. 632–640.
arXiv:cs.DS/9911003.
J. Graph Algorithms and Applications 3 (3): 1–27, 1999.Uses an idea of Baker to cover a planar graph with subgraphs of low treewidth. As a consequence, any fixed pattern can be found as a subgraph in linear time; the same methods can be used to solve other planar graph problems including vertex connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths. A companion paper, "Diameter and treewidth in minor-closed graph families", presents a result announced in the conference version of this paper, that exactly characterizes the minor-closed graph families for which the same techniques apply.
- Diameter and treewidth in minor-closed graph families.
D. Eppstein.
arXiv:math.CO/9907126.
Algorithmica 27: 275–291, 2000 (special issue on treewidth, graph minors, and algorithms).This paper introduces the diameter-treewidth property (later known as bounded local treewidth): a functional relationship between the diameter of its graph and its treewidth. Previously known results imply that planar graphs have bounded local treewidth; we characterize the other minor-closed families with this property. Specifically, minor-closed family F has bounded local treewidth if and only if there exists an apex graph G that is not in F; an apex graph is a graph that can be made planar by removing a single vertex. The minor-free families that exclude an apex graph (and therefore have bounded local treewidth) include the bounded-genus graphs (for which, as with planar graphs, we show a linear bound for the treewidth as a function of the diameter) and K3,a-free graphs. As a consequence, subgraph isomorphism for subgraphs of bounded size and approximations to several NP-hard optimization problems can be computed efficiently on these graphs, extending previous results for planar graphs.
Some of these results were announced in the conference version of "subgraph isomorphism for planar graphs and related problems" but not included in the journal version. Since its publication, there have been many more works on local treewidth. The class of problems that could be solved quickly on graphs of bounded local treewidth was extended and classified by Frick and Grohe, "Deciding first-order properties of locally tree-decomposable structures", J. ACM 48: 1184–1206, 2001; the proof that bounded local treewidth is equivalent to having an excluded apex minor was simplified, and the dependence of the treewidth on diameter improved, by a subsequent paper of Demaine and Hajiaghayi, "Diameter and treewidth in minor-closed graph families, revisited", Algorithmica 40: 211–215, 2004. The concept of local treewidth is the basis for the theory of bidimensionality, a general framework for fixed-parameter-tractable algorithms and approximation algorithms in minor-closed graph families; for a survey, see Demaine and Hajiaghayi, "The bidimensionality theory and its algorithmic applications", The Computer J. 51: 292–302, 2008.
- All maximal independent sets and dynamic dominance for sparse
graphs.
D. Eppstein.
arXiv:cs.DS/0407036.
16th ACM-SIAM Symp. Discrete Algorithms, Vancouver, 2005, pp. 451–459.
ACM Trans. Algorithms 5(4):A38, 2009.We show how to apply reverse search to list all maximal independent sets in bounded-degree graphs in constant time per set, in graphs from minor closed families in linear time per set, and in sparse graphs in subquadratic time per set. The latter two results rely on new data structures for maintaining a dynamic vertex set in a graph and quickly testing whether the set dominates all other vertices.
- Finding large clique minors is hard.
D. Eppstein.
arXiv:0807.0007.
J. Graph Algorithms and Applications 13 (2): 197–204, 2009.Proves that it's NP-complete to compute the Hadwiger number of a graph.
- Flows in one-crossing-minor-free graphs.
E. Chambers and D. Eppstein.
arXiv:1007.1484.
Proc. 21st International Symposium on Algorithms and Computation (ISAAC 2010), Jeju, Korea, 2010.
Springer, Lecture Notes in Comp. Sci. 6506, 2010, pp. 241–252.
J. Graph Algorithms and Applications 17 (3): 201–220, 2013.We show that the maximum flow problem can be solved in near-linear time for K5-minor-free and K3,3-minor-free graphs. The same result also holds for H-minor-free graphs when H can be embedded in the plane with one crossing and a structural decomposition of the input flow graph is given.
- Densities of minor-closed graph families.
D. Eppstein.
arXiv:1009.5633.
Electronic J. Combinatorics 17(1), Paper R136, 2010.For every minor-closed graph family (such as the family of planar graphs), there is a constant c such that the maximum number of edges in an n-vertex graph in the family is c(n + o(n); for instance, for planar graphs, c = 3. We call c the limiting density of the family, and we study the set of all limiting densities of all minor-closed graph families. We show that this set of limiting densities is closed and well-ordered, with order type at least ωω, and we find an exact description of the members of this set up to its first two cluster points 1 and 3/2.
- Grid minors in damaged grids.
D. Eppstein.
arXiv:1303.1136.
Electronic J. Combinatorics 21(3), Paper P3.20, 2014.We give tight bounds on the size of the largest remaining grid minor in a grid graph from which a given number of vertices have been deleted, and study several related problems.
- NC algorithms for perfect matching and maximum flow in
one-crossing-minor-free graphs.
D. Eppstein and V. V. Vazirani.
arXiv:1802.00084.
Proc. 30th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2019), Phoenix, Arizona, 2019, pp. 23–30.
SIAM J. Computing 50 (3): 1014–1033, 2021.We extend Anari and Vazirani's parallel algorithm for perfect matching in planar graphs to the graph families with a forbidden minor with crossing number one, by developing a concept of mimicking networks for perfect matching.
(Slides)
- Minor-closed graph classes with bounded layered pathwidth.
V. Dujmović, D. Eppstein, G. Joret, P. Morin, and D. R. Wood.
arXiv:1810.08314.
SIAM J. Discrete Math 34 (3): 1693–1709, 2020.A minor-closed graph family has a functional relation between diameter and path width (bounded local pathwidth) if and only if it excludes an apex-tree. The same graph families are also the ones with bounded layered pathwidth: a simultaneous path decomposition and layering (sequence of subsets of vertices with all edges connecting the same subset or consecutive subsets) so that the intersection of a bag and a layer has constant size.
- Homotopy height, grid-minor height and graph-drawing height.
T. Biedl, E. Chambers, D. Eppstein, A. de Mesmay, and T. Ophelders.
arXiv:1908.05706.
Proc. 27th Int. Symp. Graph Drawing, Prague, Czech Republic, 2019.
Springer, Lecture Notes in Comp. Sci. 11904 (2019), pp. 468–481.
We lower bound the height of a drawing of a planar graph in an integer grid by a topological invariant, the homotopy height, and show that this invariant is equal to the smallest height of a grid graph in which the given graph is a minor.
- Product structure extension of the Alon–Seymour–Thomas theorem.
M. Distel, V. Dujmović, D. Eppstein, R. Robert Hickingbotham, G. Joret, P. Micek, P. Morin, M. T. Seweryn, and D. R. Wood.
arXiv:2212.08739.
SIAM J. Discrete Math. 38 (3): 2095–2107, 2024.
The graphs in any nontrivial minor-closed graph family can be represented as strong products of a graph of treewidth 4 with a clique of size \(O(\sqrt{n})\). For planar graphs and \(K_{3,t}\)-minor-free graphs, the treewidth can be reduced to 2.