One-factorizations of regular graphs of order 12.
On-line 𝓟-coloring of graphs
For a given induced hereditary property 𝓟, a 𝓟-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property 𝓟. We consider the effectiveness of on-line 𝓟-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound for the performance guarantee function of any stingy on-line 𝓟-coloring algorithm. In the class of generalized...
Operations on directed graphs.
Optimal double-loop networks with non-unit steps.
Optimal edge ranking of complete bipartite graphs in polynomial time
An edge ranking of a graph is a labeling of edges using positive integers such that all paths connecting two edges with the same label visit an intermediate edge with a higher label. An edge ranking of a graph is optimal if the number of labels used is minimum among all edge rankings. As the problem of finding optimal edge rankings for general graphs is NP-hard [12], it is interesting to concentrate on special classes of graphs and find optimal edge rankings for them efficiently. Apart from trees...
Parallel Algorithms for Maximal Cliques in Circle Graphs and Unrestricted Depth Search
We present parallel algorithms on the BSP/CGM model, with p processors, to count and generate all the maximal cliques of a circle graph with n vertices and m edges. To count the number of all the maximal cliques, without actually generating them, our algorithm requires O(log p) communication rounds with O(nm/p) local computation time. We also present an algorithm to generate the first maximal clique in O(log p) communication rounds with O(nm/p) local computation, and to generate each one of...
Parallel algorithms on interval graphs
Partial covers of graphs
Given graphs G and H, a mapping f:V(G) → V(H) is a homomorphism if (f(u),f(v)) is an edge of H for every edge (u,v) of G. In this paper, we initiate the study of computational complexity of locally injective homomorphisms called partial covers of graphs. We motivate the study of partial covers by showing a correspondence to generalized (2,1)-colorings of graphs, the notion stemming from a practical problem of assigning frequencies to transmitters without interference. We compare the problems of...
Perfect matching in general vs. cubic graphs : a note on the planar and bipartite cases
Perfect Matching in General vs. Cubic Graphs: A Note on the Planar and Bipartite Cases
It is known that finding a perfect matching in a general graph is AC0-equivalent to finding a perfect matching in a 3-regular (i.e. cubic) graph. In this paper we extend this result to both, planar and bipartite cases. In particular we prove that the construction problem for perfect matchings in planar graphs is as difficult as in the case of planar cubic graphs like it is known to be the case for the famous Map Four-Coloring problem. Moreover we prove that the existence and construction...
Perfectly matchable subgraph problem on a bipartite graph
We consider the maximum weight perfectly matchable subgraph problem on a bipartite graph G=(UV,E) with respect to given nonnegative weights of its edges. We show that G has a perfect matching if and only if some vector indexed by the nodes in UV is a base of an extended polymatroid associated with a submodular function defined on the subsets of UV. The dual problem of the separation problem for the extended polymatroid is transformed to the special maximum flow problem on G. In this paper, we give...
Permanents, Pfaffian orientations, and even directed circuits.
Planar graphs with topological constraints.
Polynomial time algorithms for two classes of subgraph problem
We design a O(n3) polynomial time algorithm for finding a (k-1)- regular subgraph in a k-regular graph without any induced star K1,3(claw-free). A polynomial time algorithm for finding a cubic subgraph in a 4-regular locally connected graph is also given. A family of k-regular graphs with an induced star K1,3 (k even, k ≥ 6), not containing any (k-1)-regular subgraph is also constructed.
Polynomial-time partition of a graph into cliques.
Precoloring extension with fixed color bound.
Primal-dual approximation algorithms for a packing-covering pair of problems
We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph. The problem pair involves weights and capacities; we consider the case of unit weights and the case of unit capacities. In each case we describe a simple algorithm that outputs a solution to the packing problem and...
Primal-dual approximation algorithms for a packing-covering pair of problems
We consider a special packing-covering pair of problems. The packing problem is a natural generalization of finding a (weighted) maximum independent set in an interval graph, the covering problem generalizes the problem of finding a (weighted) minimum clique cover in an interval graph. The problem pair involves weights and capacities; we consider the case of unit weights and the case of unit capacities. In each case we describe a simple algorithm that outputs a solution to the packing problem and...
Radial level planarity testing and embedding in linear time.
Random sampling of labeled tournaments.