299 research outputs found
Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms
Given a graph H = (V_H,E_H) and a positive integer k, the k-th power of H, written H^k, is the graph obtained from H by adding edges between any pair of vertices at distance at most k in H; formally, H^k = (V_H, {xy | 1 <= d_H (x, y) <= k}). A graph G is the k-th power of a graph H if G = H^k, and in this case, H is a k-th root of G. Our investigations deal with the computational complexity of recognizing k-th powers of general graphs as well as restricted graphs. This work provides new NP-completeness results, good characterizations and efficient algorithms for graph powers
Computing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if
and only if x, y are of distance at most two in H. Given H it is easy to
compute its square H2, however Motwani and Sudan proved that it is NP-complete
to determine if a given graph G is the square of some graph H (of girth 3). In
this paper we consider the characterization and recognition problems of graphs
that are squares of graphs of small girth, i.e. to determine if G = H2 for some
graph H of small girth. The main results are the following. - There is a graph
theoretical characterization for graphs that are squares of some graph of girth
at least 7. A corollary is that if a graph G has a square root H of girth at
least 7 then H is unique up to isomorphism. - There is a polynomial time
algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is
NP-complete to recognize if G = H2 for some graph H of girth 4. These results
almost provide a dichotomy theorem for the complexity of the recognition
problem in terms of girth of the square roots. The algorithmic and graph
theoretical results generalize previous results on tree square roots, and
provide polynomial time algorithms to compute a graph square root of small
girth if it exists. Some open questions and conjectures will also be discussed
Sparse square roots.
We show that it can be decided in polynomial time whether a graph of maximum degree 6 has a square root; if a square root exists, then our algorithm finds one with minimum number of edges. We also show that it is FPT to decide whether a connected n-vertex graph has a square root with at most n − 1 + k edges when this problem is parameterized by k. Finally, we give an exact exponential time algorithm for the problem of finding a square root with maximum number of edges
The square of a block graph
AbstractThe square H2 of a graph H is obtained from H by adding new edges between every two vertices having distance two in H. A block graph is one in which every block is a clique. For the first time, good characterizations and a linear time recognition of squares of block graphs are given in this paper. Our results generalize several previous known results on squares of trees
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Constrained Representations of Map Graphs and Half-Squares
The square of a graph H, denoted H^2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B=(X,Y,E_B) are the subgraphs of B^2 induced by the color classes X and Y, B^2[X] and B^2[Y]. For a given graph G=(V,E_G), if G=B^2[V] for some bipartite graph B=(V,W,E_B), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM49 (2) (2002) 127-138].
While Chen, Grigni, Papadimitriou proved that any map graph G=(V,E_G) has a witness with at most 3|V|-6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown.
We also consider half-squares of tree-convex bipartite graphs and prove the following complexity dichotomy: Given a graph G=(V,E_G) and an integer k, deciding if G=B^2[V] for some tree-convex bipartite graph B=(V,W,E_B) with |W|<=k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs
EDGE IDEALS OF SQUARES OF TREES
We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees T for which the square is co-chordal, that is the complement of the square, (T2)c, is a chordal graph. For particular classes of trees such as paths and double brooms, we determine the Krull dimension and the projective dimension
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