Rogers Mathew

Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

We present two online algorithms for maintaining a topological order of a directed <i>n</i>-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles <i>m</i> arc additions in O(<i>m</i>^3/2) time. For sparse graphs (<i>m</i>/<i>n</i> = O(1)), this bound improves the best previous bound by a logarithmic factor, and is tight to within a constant factor among algorithms satisfying a natural locality property. Our second algorithm handles an arbitrary sequence of arc additions in O(<i>n</i>^5/2) time. For sufficiently dense graphs, this bound improves the best previous bound by a polynomial factor. Our bound may be far from tight: we show that the algorithm can take Ω(<i>n</i>²2^{√2 lg} <i>n</i>) time by relating its performance to a generalization of the <i>k</i>-levels problem of combinatorial geometry. A completely different algorithm running in Θ(<i>n</i>² log <i>n</i>) time was given recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to the maintenance of strong components, without affecting the asymptotic time bounds.

Faster Algorithms for Incremental Topological Ordering

We present two online algorithms for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm takes O(m1/2) amortized time per arc and our second algorithm takes O(n2.5/m) amortized time per arc, where nis the number of vertices and mis the total number of arcs. For sparse graphs, our O(m1/2) bound improves the best previous bound by a factor of lognand is tight to within a constant factor for a natural class of algorithms that includes all the existing ones. Our main insight is that the two-way search method of previous algorithms does not require an ordered search, but can be more general, allowing us to avoid the use of heaps (priority queues). Instead, the deterministic version of our algorithm uses (approximate) median-finding; the randomized version of our algorithm uses uniform random sampling. For dense graphs, our O(n2.5/m) bound improves the best previously published bound by a factor of n1/4and a recent bound obtained independently of our work by a factor of logn. Our main insight is that graph search is wasteful when the graph is dense and can be avoided by searching the topological order space instead. Our algorithms extend to the maintenance of strong components, in the same asymptotic time bounds.

Chordal Bipartite Graphs with High Boxicity

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.

Boxicity of line graphs

The boxicity of a graph H , denoted by box ( H ) , is the minimum integer k such that H is an intersection graph of axis-parallel k -dimensional boxes in R k . In this paper we show that for a line graph G of a multigraph, box ( G ) ? 2 Δ ( G ) ( ? log 2 log 2 Δ ( G ) ? + 3 ) + 1 , where Δ ( G ) denotes the maximum degree of G . Since G is a line graph, Δ ( G ) ? 2 ( ? ( G ) - 1 ) , where ? ( G ) denotes the chromatic number of G , and therefore, box ( G ) = O ( ? ( G ) log 2 log 2 ( ? ( G ) ) ) . For the d -dimensional hypercube Q d , we prove that box ( Q d ) ? 1 2 ( ? log 2 log 2 d ? + 1 ) . The question of finding a nontrivial lower bound for box ( Q d ) was left open by Chandran and Sivadasan in L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795-5800].The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once). Highlights? Boxicity of a line graph G is O ( D ( log log D ) ) , where D denotes the maximum degree of G . ? Boxicity of a line graph G is O ( k ( log log k ) ) , where k denotes the chromatic number of G . ? Boxicity of the d -dimensional hypercube is at least ( 1 / 2 ) ( ? log log d ? + 1 ) .

SEPARATION DIMENSION OF BOUNDED DEGREE GRAPHS

The separation dimension of a graph

A Dirac-type characterization of k-chordal graphs

G

Cubicity, Degeneracy, and Crossing Number

is the smallest natural number

Separation Dimension of Graphs and Hypergraphs

k

Maximum Weighted Independent Sets with a Budget

for which the vertices of

Induced Separation Dimension

G