Tree decomposition of graphs

Let H be a tree on h≥2 vertices. It is shown that if G=(V, E) is a graph with \delta (G)\ge (|V|/2)+10h^4\sqrt{|V|\log|V|} , and h−1 divides |E|, then there is a decomposition of the edges of G into copies of H. This result is asymptotically the best possible for all trees with at least three vertices. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 237–251, 1998     

Tree spanners in planar graphs

A tree t-spanner of a graph G is a spanning subtree T of G in which the distance between every pair of vertices is at most t times their distance in G. Spanner problems have received some attention, mostly in the context of communication networks. It is known that for general unweighted graphs, the problem of deciding the existence of a tree t-spanner can be solved in polynomial time for t=2, while it is NP-hard for any t⩾4; the case t=3 is open, but has been conjectured to be hard. In this paper, we consider tree spanners in planar graphs. We show that even for planar unweighted graphs, it is NP-hard to determine the minimum t for which a tree t-spanner exists. On the other hand, we give a polynomial algorithm for any fixed t that decides for planar unweighted graphs with bounded face length whether there is a tree t-spanner. Furthermore, we prove that it can be decided in polynomial time whether a planar unweighted graph has a tree t-spanner for t=3.     

Tree decompositions of graphs without large bipartite holes

A recent result of Condon, Kim, Kühn, and Osthus implies that for any , an n‐vertex almost r‐regular graph G has an approximate decomposition into any collections of n‐vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost αn‐regular graph G with any α>0 and a collection of bounded degree trees on at most (1?o(1))n vertices if G does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into n‐vertex trees. Moreover, this implies that for any α>0 and an n‐vertex almost αn‐regular graph G, with high probability, the randomly perturbed graph has an approximate decomposition into all collections of bounded degree trees of size at most (1?o(1))n simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey‐Turán theory and the randomly perturbed graph model.     

Backbone colorings for graphs: Tree and path backbones

We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,…} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly . We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer ?, is there a backbone coloring for G and T with at most ? colors?” jumps from polynomial to NP‐complete between ? = 4 (easy for all spanning trees) and ? = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 137–152, 2007     

Tree components in random distance graphs of special form

In this work, we consider the problem of distribution of the number of tree components with given number of vertices k(k ?? 2) for a certain series of random distance graphs. Generalizations of the classical Erd?s?CRényi results are obtained in the case of geometric graphs of special form.     

Tree coloring of distance graphs with a real interval set

Let R be the set of real numbers and D be a subset of the positive real numbers. The distance graph G(R,D) is a graph with the vertex set R and two vertices x and y are adjacent if and only if |x−y|D. In this work, the vertex arboricity (i.e., the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces an acyclic subgraph) of G(R,D) is determined for D being an interval between 1 and δ.     

Tree decompositions of graphs: Saving memory in dynamic programming

We propose a simple and effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique” is based on a tree-like set covering problem. We substantiate our findings by experimental results. Our strategy has negligible computational overhead concerning running time but achieves memory savings for nice tree decompositions and path decompositions between 60% and 98%.     

Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms

A spanning tree T of a graph G is said to be a treet-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4 and is linearly solvable for t≤2. The case t=3 still remains open. A chordal graph is called a 2-sep chordal graph if all of its minimal a−b vertex separators for every pair of non-adjacent vertices a and b are of size two. It is known that not all 2-sep chordal graphs admit tree 3-spanners. This paper presents a structural characterization and a linear time recognition algorithm of tree 3-spanner admissible 2-sep chordal graphs. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep chordal graph is proposed.     

Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction

A spanning tree T of a graph G is called a treet-spanner, if the distance between any two vertices in T is at most t-times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4, and is linearly solvable for t=1 and t=2. The case t=3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a−b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81-87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203-210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.     

Graph equations for line graphs,total graphs,middle graphs and quasi-total graphs

Let G be a graph with vertex-set V(G) and edge-set X(G). Let L(G) and T(G) denote the line graph and total graph of G. The middle graph M(G) of G is an intersection graph Ω(F) on the vertex-set V(G) of any graph G. Let F = V′(G) ∪ X(G) where V′(G) indicates the family of all one-point subsets of the set V(G), then $\text{M(G) = Ω(F)}$.The quasi-total graph P(G) of G is a graph with vertex-set V(G)∪X(G) and two vertices are adjacent if and only if they correspond to two non-adjacent vertices of G or to two adjacent edges of G or to a vertex and an edge incident to it in G.In this paper we solve graph equations $\text{L(G) ? P(H);}\text{L(G)}\text{? P(H); P(G) ? T(H);}\text{P(G)}\text{? T(H); M(G) ? P(H);}\text{M(G)}\text{? P(H)}$.     

Comparability graphs and intersection graphs

A function diagram (f-diagram) D consists of the family of curves {$\text{1}\text{?}$ñ} obtained from n continuous functions $\text{f}{}_{\mathrm{i}}\text{:[0,1]→}\text{R}\text{(1?i?n)}$. We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement ? is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the f_i are linear functions. It is also shown that G is the intersection graph of the concatenation of ?k permutation diagrams if and only if the partial order dimension of $\text{G}\text{?}\text{is ?k+1}$. Computational complexity results are obtained for recognizing such graphs.     

Clique graphs of time graphs

A simple, finite graph G is called a time graph (equivalently, an indifference graph) if there is an injective real function f on the vertices v(G) such that v_iv_j ∈ e(G) for v_i ≠ v_j if and only if |f(v_i) ? f(v_j)| ≤ 1. A clique of a graph G is a maximal complete subgraph of G. The clique graph K(G) of a graph G is the intersection graph of the cliques of G. It will be shown that the clique graph of a time graph is a time graph, and that every time graph is the clique graph of some time graph. Denote the clique graph of a clique graph of G by K²(G), and inductively, denote K(K^m?1(G)) by K^m(G). Define the index indx(G) of a connected time graph G as the smallest integer n such that Kⁿ(G) is the trivial graph. It will be shown that the index of a time graph is equal to its diameter. Finally, bounds on the diameter of a time graph will be derived.     

Cover-incomparability graphs and chordal graphs

The problem of recognizing cover-incomparability graphs (i.e. the graphs obtained from posets as the edge-union of their covering and incomparability graph) was shown to be NP-complete in general [J. Maxová, P. Pavlíkova, A. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229-236], while it is for instance clearly polynomial within trees. In this paper we concentrate on (classes of) chordal graphs, and show that any cover-incomparability graph that is a chordal graph is an interval graph. We characterize the posets whose cover-incomparability graph is a block graph, and a split graph, respectively, and also characterize the cover-incomparability graphs among block and split graphs, respectively. The latter characterizations yield linear time algorithms for the recognition of block and split graphs, respectively, that are cover-incomparability graphs.     

Embedding graphs in Cayley graphs

IfY is a finite graph then it is known that every sufficiently large groupG has a Cayley graph containing an induced subgraph isomorphic toY. This raises the question as to what is sufficiently large. Babai and Sós have used a probabilistic argument to show that |G| > 9.5 |Y|³ suffices. Using a form of greedy algorithm we strengthen this to (2 + \sqrt 3 )|Y|^3 $$ " align="middle" border="0"> . Some related results on finite and infinite groups are included.     

Universal graphs and induced-universal graphs

We construct graphs that contain all bounded-degree trees on n vertices as induced subgraphs and have only cn edges for some constant c depending only on the maximum degree. In general, we consider the problem of determining the graphs, so-called universal graphs (or induced-universal graphs), with as few vertices and edges as possible having the property that all graphs in a specified family are contained as subgraphs (or induced subgraphs). We obtain bounds for the size of universal and induced-universal graphs for many classes of graphs such as trees and planar graphs. These bounds are obtained by establishing relationships between the universal graphs and the induced-universal graphs.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non-bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.