Search

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP , V P , and VNP . We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively. Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials. [ABSTRACT FROM AUTHOR]

After the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover Number . Here we consider the treewidth and pathwidth problems parameterized by k , the size of a minimum vertex cover of the input graph. We show that the pathwidth and treewidth can be computed in O ∗ ( 3 k ) time. This complements recent polynomial kernel results for treewidth and pathwidth parameterized by the Vertex Cover Number . [ABSTRACT FROM AUTHOR]

Non-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q ≥ 0 , the q -limited search number, s q ( G ) , of a graph G is the smallest number of searchers required to capture an invisible fugitive in G , when the searchers are allowed to know the position of the fugitive at most q times. The search parameter s 0 ( G ) corresponds to the pathwidth of a graph G , and s ∞ ( G ) to its treewidth. Determining s q ( G ) is NP-complete for any fixed q ≥ 0 in general graphs and s 0 ( T ) can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any q > 0 . We introduce a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by rs q and is shown to be equal to the non-deterministic graph searching parameter s q for q = 0 , 1 , and at most twice s q for any q ≥ 2 (for any graph G ). Our main result is a polynomial time algorithm that computes rs q ( T ) for any tree T and any q ≥ 0 . This provides a 2-approximation of s q ( T ) for any tree T , and shows that the decision problem associated to s 1 is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest. [ABSTRACT FROM AUTHOR]

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth- $$2$$ outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $$2n$$ vertices are bounded by $$O(n^{3})$$ and $$O(n^{10})$$ , in the convex and general case, respectively. We then apply similar methods to prove an $$n^{O(\log (n))}$$ upper bound on the Ramsey number of a path with $$n$$ ordered vertices. [ABSTRACT FROM AUTHOR]

Abstract: The problems of (classical) searching and connected searching of weighted trees are known to be computationally hard. In this work we give a polynomial-time 3-approximation algorithm that finds a connected search strategy of a given weighted tree. This in particular yields constant factor approximation algorithms for the (non-connected) classical searching problems and for the weighted pathwidth problem for this class of graphs. [Copyright &y& Elsevier]

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in monadic second-order logic with second-order quantification on vertex sets, which includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless P = NP. We also show that, given a graph G and an integer k, deciding whether the clique-width of G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s. [ABSTRACT FROM AUTHOR]

Let P(G; x; y) be the number of vertex colorings &phi : V → {1,...,x} of an undirected graph G = (V, E) such that for all edges { u, v} ≠ E the relations &phi(v) ≤ y and φ(v) ≤ y implyφ(u) ≠ φ(v). We show that P(G; x, y) is a polynomial in x and y which is closely related to Stanley's chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of G. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex, and one in terms of the lattice of forbidden colorings. Finally, we give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that P(G; x, y) can be evaluated in polynomial time for trees and graphs of restricted pathwidth. [ABSTRACT FROM AUTHOR]