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1.
An axis-parallel k-dimensional box is a Cartesian product R 1×R 2×???×R k where R i (for 1≤i≤k) is a closed interval of the form [a i ,b i ] on the real line. For a graph G, its boxicity box?(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}An axis-parallel k-dimensional box is a Cartesian product R
1×R
2×⋅⋅⋅×R
k
where R
i
(for 1≤i≤k) is a closed interval of the form [a
i
,b
i
] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc.
A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs.
For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set
problem for boxicity d graphs, given a box representation, has a
?1+\frac1clogn?d-1\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}
approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether
the boxicity of a graph is at most 2 itself is NP-hard. 相似文献
2.
《国际计算机数学杂志》2012,89(9):1897-1910
In this paper we obtain information about the hyperbolicity constant of cubic graphs. They are a very interesting class of graphs with many applications; furthermore, they are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants, i.e. the graphs which are like trees (in the Gromov sense). Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K4 or to K3, 3, the inequalities 5k/4≤δ (?)≤3k/2 hold, if k is the length of every edge in G. 相似文献
3.
Given a pattern graph H with l edges, and a host graph G guaranteed to contain at most one occurrence of a subgraph isomorphic to H, we show that the time complexity of the problem of finding such an occurrence (if any) in G as well as that of the decision version of the problem are within a multiplicative factor O(l) of the time complexity for the corresponding problem in the general case, when G may contain several occurrences of H. It follows that for pattern graphs of constant size, the aforementioned uniqueness guarantee cannot yield any asymptotic speed up. We also derive analogous results with the analogous multiplicative factor linear in the number of vertices of H in the induced case when occurrences of induced subgraphs of G isomorphic to H are sought. 相似文献
4.
In a graph, a vertex is simplicial if its neighborhood is a clique. For an integer k≥1, a graph G=(VG,EG) is the k-simplicial power of a graph H=(VH,EH) (H a root graph of G) if VG is the set of all simplicial vertices of H, and for all distinct vertices x and y in VG, xyEG if and only if the distance in H between x and y is at most k. This concept generalizes k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees; k-leaf powers are the k-simplicial powers of trees. Recently, a lot of work has been done on k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For k≤5, k-leaf powers can be recognized in linear time, and for k≤4, structural characterizations are known. For k≥6, the recognition and characterization problems of k-leaf powers are still open. Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related, and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs. 相似文献
5.
The universal cover T
G
of a connected graph G is the unique (possibly infinite) tree covering G, i.e., that allows a locally bijective homomorphism from T
G
to G. It is well-known that if a graph G covers a graph H, then their universal covers are isomorphic, and that the latter can be tested in polynomial time by checking if G and H share the same degree refinement matrix. We extend this result to locally injective and locally surjective homomorphisms
by following a very different approach. Using linear programming techniques we design two polynomial time algorithms that
check if there exists a locally injective or a locally surjective homomorphism, respectively, from a universal cover T
G
to a universal cover T
H
(both given by their degree matrices). This way we obtain two heuristics for testing the corresponding locally constrained
graph homomorphisms. Our algorithm can also be used for testing (subgraph) isomorphism between universal covers, and for checking
if there exists a locally injective or locally surjective homomorphism (role assignment) from a given tree to an arbitrary
graph H. 相似文献
6.
Continuous-time quantum walk (CTQW) over finite group schemes is investigated, where it is shown that some properties of a
CTQW over a group scheme defined on a finite group G induces a CTQW over group scheme defined on G/H, where H is a normal subgroup of G with prime index. This reduction can be helpful in analyzing CTQW on underlying graphs of group schemes. Even though this
claim is proved for normal subgroups with prime index (using the Clifford’s theorem from representation theory), it is checked
in some examples that for other normal subgroups or even non-normal subgroups, the result is also true! Moreover, it is shown
that the Bose-Mesner (BM) algebra associated with the group scheme over G is isomorphic to the corresponding BM algebra of the association scheme defined over the coset space G/H up to the scale factor |H|. In fact, we show that the underlying graph defined over group G is a covering space for the quotient graph defined over G/H, so that CTQW over the graph on G, starting from any arbitrary vertex, is isomorphic to the CTQW over the quotient graph on G/H if we take the sum of the amplitudes corresponding to the vertices belonging to the same cosets. 相似文献
7.
Feodor F. Dragan Fedor V. Fomin Petr A. Golovach 《Journal of Computer and System Sciences》2011,77(6):1108-1119
A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t?3 and is NP-complete when t is part of the input. They also left as an open problem if the problem is polynomial time solvable for every fixed t?4. In this work we resolve the open question of Fekete and Kremer by proving much more general results:
•
The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear. •
Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus. •
Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t?4, the problem of finding a tree t-spanner is NP-complete on K6-minor-free graphs.
8.
The minimum k-terminal cut problem is of considerable theoretical interest and arises in several
applied areas such as parallel and distributed computing, VLSI circuit design, and networking. In this paper
we present two new approximation and exact algorithms for this problem on an n-vertex undirected weighted
planar graph G. For the case when the k terminals are covered by the boundaries of m > 1 faces of G, we
give a min{O(n
2 log n logm), O(m
2
n
1.5 log2
n + k
n)} time algorithm with a (2–2/k)-approximation ratio
(clearly, m \le k). For the case when all k terminals are covered by the boundary of one face of G, we give an
O(n
k3 + (n log n)k
2) time exact algorithm, or a linear time exact algorithm if k = 3, for computing an optimal
k-terminal cut. Our algorithms are based on interesting observations and improve the previous algorithms when
they are applied to planar graphs. To our best knowledge, no previous approximation algorithms specifically
for solving the k-terminal cut problem on planar graphs were known before. The (2–2/k)-approximation
algorithm of Dahlhaus et al. (for general graphs) takes O(k
n
2 log n) time when applied to planar graphs. Our
approximation algorithm for planar graphs runs faster than that of Dahlhaus et al. by at least an O(k/logm)
factor (m \le k). 相似文献
9.
The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an algorithm to obtain an outerplanar subgraph of a graph with disk dimension k by removing at most 2k−2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms on graphs with fixed disk dimension. In particular, a linear-time approximation algorithm is presented for the pathwidth problem. 相似文献
10.
The minimum k-terminal cut problem is of considerable theoretical interest and arises in several
applied areas such as parallel and distributed computing, VLSI circuit design, and networking. In this paper
we present two new approximation and exact algorithms for this problem on an n-vertex undirected weighted
planar graph G. For the case when the k terminals are covered by the boundaries of m > 1 faces of G, we
give a min{O(n
2 log n logm), O(m
2
n
1.5 log2
n + k
n)} time algorithm with a (2–2/k)-approximation ratio
(clearly, m \le k). For the case when all k terminals are covered by the boundary of one face of G, we give an
O(n
k3 + (n log n)k
2) time exact algorithm, or a linear time exact algorithm if k = 3, for computing an optimal
k-terminal cut. Our algorithms are based on interesting observations and improve the previous algorithms when
they are applied to planar graphs. To our best knowledge, no previous approximation algorithms specifically
for solving the k-terminal cut problem on planar graphs were known before. The (2–2/k)-approximation
algorithm of Dahlhaus et al. (for general graphs) takes O(k
n
2 log n) time when applied to planar graphs. Our
approximation algorithm for planar graphs runs faster than that of Dahlhaus et al. by at least an O(k/logm)
factor (m \le k). 相似文献
11.
We shall present an algorithm for determining whether or not a given planar graph H can ever be a subgraph of a 4-regular planar graph. The algorithm has running time O(|H|2.5) and can be used to find an explicit 4-regular planar graph G⊃H if such a graph exists. It shall not matter whether we specify that H and G must be simple graphs or allow them to be multigraphs. 相似文献
12.
《国际计算机数学杂志》2012,89(5):905-915
A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbours in S. The k-domination number γ k (G) is the minimum cardinality of a k-dominating set of G, and α(G) denotes the cardinality of a maximum independent set of G. Brook's well-known bound for the chromatic number χ and the inequality α(G)≥n(G)/χ(G) for a graph G imply that α(G)≥n(G)/Δ(G) when G is non-regular and α(G)≥n(G)/(Δ(G)+1) otherwise. In this paper, we present a new proof of this property and derive some bounds on γ k (G). In particular, we show that, if G is connected with δ(G)≥k then γ k (G)≤(Δ(G)?1)α(G) with the exception of G being a cycle of odd length or the complete graph of order k+1. Finally, we characterize the connected non-regular graphs G satisfying equality in these bounds and present a conjecture for the regular case. 相似文献
13.
Output-Sensitive Reporting of Disjoint Paths 总被引:1,自引:0,他引:1
A k -path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper we study the problem of
performing k -path queries, with , in a graph G with n vertices. We denote with the total length of the reported paths. For , we present an optimal data structure for G that uses O(n) space and executes k -path queries in output-sensitive time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as
bipolar (st ) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex
grid drawings of triconnected planar graphs.
Received August 24, 1996; revised April 8, 1997. 相似文献
14.
In automatic graph drawing a given graph has to be laid out in the plane, usually according to a number of topological and aesthetic constraints. Nice drawings for sparse nonplanar graphs can be achieved by determining a maximum planar subgraph and augmenting an embedding of this graph. This approach appears to be of limited value in practice, because the maximum planar subgraph problem is NP-hard.We attack the maximum planar subgraph problem with a branch-and-cut technique which gives us quite good, and in many cases provably optimum, solutions for sparse graphs and very dense graphs. In the theoretical part of the paper, the polytope of all planar subgraphs of a graphG is defined and studied. All subgraphs of a graphG, which are subdivisions ofK
5 orK
3,3, turn out to define facets of this polytope. For cliques contained inG, the Euler inequalities turn out to be facet-defining for the planar subgraph polytope. Moreover, we introduce the subdivision inequalities,V
2k
inequalities, and the flower inequalities, all of which are facet-defining for the polytope. Furthermore, the composition of inequalities by 2-sums is investigated.We also present computational experience with a branch-and-cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision ofK
5 orK
3,3. These structures give us inequalities which are used as cutting planes.Finally, we try to convince the reader that the computation of maximum planar subgraphs is indeed a practical tool for finding nice embeddings by applying this method to graphs taken from the literature. 相似文献
15.
Takayuki Nagoya 《Theory of Computing Systems》2009,44(3):356-368
It is known that, given two isomorphic graphs G and H, finding a pair of vertices (v
i
,w
j
) where v
i
is mapped to w
j
by an isomorphism from G to H is as hard as computing an isomorphism from G to H. In this paper, we prove a similar result for the Graph Automorphism problem. That is to say, we prove that, given a graph
that has a non-trivial automorphism, finding a pair of vertices (v
i
,v
j
) where v
i
is mapped to v
j
by a non-trivial automorphism on the graph is as hard as computing a non-trivial automorphism on the graph. 相似文献
16.
《国际计算机数学杂志》2012,89(10):2202-2211
Let G be a graph, and let a, b, k be integers with 0≤a≤b, k≥0. An [a, b]-factor of graph G is defined as a spanning subgraph F of G such that a≤d F (x)≤b for each x∈V(G). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this article, a sufficient condition is given, which is a neighborhood condition for a graph G to be an (a, b, k)-critical graph. 相似文献
17.
Isolde Adler Frederic Dorn Fedor V. Fomin Ignasi Sau Dimitrios M. Thilikos 《Algorithmica》2012,64(1):69-84
Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components C u and C v . A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time $\mathcal{O}(2^{\mathcal{O}(h)} \cdot n +n^{2}\cdot\log n)$ a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming. 相似文献
18.
《国际计算机数学杂志》2012,89(4):726-732
A k-adjacent vertex distinguishing edge colouring or a k-avd-colouring of a graph G is a proper k-edge colouring of G such that no pair of adjacent vertices meets the same set of colours. The avd-chromatic number, denoted by χ′a(G), is the minimum number of colours needed in an avd-colouring of G. It is proved that for any connected 3-colourable Hamiltonian graph G, we have χ′a(G)≤Δ+3. 相似文献
19.
《国际计算机数学杂志》2012,89(3):522-526
A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ u∈V f(u). The Roman domination number γR(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γR(H)/3. 相似文献
20.
Given a graph G=(V,E), the well-known spanning forest problem of G can be viewed as the problem of finding a maximal subset F of edges in G such that the subgraph induced by F is acyclic. Although this problem has well-known efficient NC algorithms, its vertex counterpart, the problem of finding a maximal subset U of vertices in G such that the subgraph induced by U is acyclic, has not been shown to be in NC (or even in RNC) and is not believed to be parallelizable in general. In this paper we present NC algorithms for solving the latter problem for two special cases. First, we show that, for a planar graph with n vertices, the problem can be solved in time with O(n) processors on an EREW PRAM. Second, we show that the problem is solvable in NC if the input graph G has only vertex-induced paths of length polylogarithmic in the number of vertices of G. As a consequence of this result, we show that certain natural extensions of the well-studied maximal independent set problem
remain solvable in NC. Moreover, we show that, for a constant-degree graph with n vertices, the problem can be solved in time with O(n
2
) processors on an EREW PRAM.
Received July 3, 1995; revised April 1, 1996. 相似文献