Gábor Bacsó

Complete description of forbidden subgraphs in the structural domination problem

A class D of graphs is concise if it only contains connected graphs and is closed under taking connected induced subgraphs. This paper is concerned with concise classes of graphs. A graph G is D-dominated if there exists a dominating subgraph D@?D in G. A connected graph G is minimal non-D-dominated if it is not D-dominated but all of its proper connected induced subgraphs are. We will give a complete description for the minimal non-D-dominated graphs for a concise D. The proof uses two stronger results.

On Minimally Imperfect Graphs with Circular Symmetry

Let G be a connected graph on n vertices. A spanning tree T of G is called an independence tree, if the set of end vertices of T (vertices with degree one in T) is an independent set in G. If G has an independence tree, then α t(G) denotes the maximum number of end vertices of an independence tree of G. We show that determining αt of a graph is an NP-hard problem. We give the following analogue of a well-known result due to Chvatal and Erdos. If αt(G) ≤ κ(G) - 1, then G is hamiltonian. We show that the condition is sharp. An I≤k-tree of G is an independence tree of G with at most k end vertices or a Hamilton cycle of G. We prove the following two generalizations of a theorem of Ore. If G has an independence tree T with k end vertices such that two end vertices of T have degree sum at least n - k + 2 in G, then G has an I≤k-1-tree. If the degree sum of each pair of nonadjacent vertices of G is at least n - k + 1, then G has an I≤k-tree. Finally, we prove the following analogue of a closure theorem due to Bondy and Chvatal. If the degree sum of two nonadjacent vertices u and v of G is at least n - 1, then G has an I≤k-tree if and only if G + uv has an I≤k-tree (k ≥ 2). The last three results are all best possible with respect to the degree sum conditions.

H-free graphs, Independent Sets, and subexponential-time algorithms

It is an outstanding open question in algorithmic graph theory to determine the complexity of the MAXIMUM INDEPENDENT SET problem on P_t-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t at most 5 [Lokshtanov et al., SODA 2014, 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t=5 [Discrete App. Math., 158 (2010), 1041-1044], we show that for any t at least 5, there is an algorithm for MAXIMUM INDEPENDENT SET on P_t-free graphs whose running time is subexponential in the number of vertices. SCATTERED SET is the generalization of MAXIMUM INDEPENDENT SET where the vertices of the solution are required to be at distance at least

Dominating bipartite subgraphs in graphs

d

On a conjecture about uniquely colorable perfect graphs

from each other. We give a complete characterization of those graphs H for which SCATTERED SET on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges): * If every component of H is a path, then d-SCATTERED SET on H-free graphs with n vertices and m edges can be solved in time 2^{(n+m)^{1-O(1/|V(H)|)}}, even if d is part of the input. * Otherwise, assuming ETH, there is no 2^{o(n+m)} time algorithm for d-SCATTERED SET for any fixed d at least 3 on H-free graphs with n vertices and m edges.

Distance domination versus iterated domination

A graph G is hereditarily dominated by a class D of connected graphs if each connected induced subgraph of G contains a dominating induced subgraph belonging to D. In this paper we characterize graphs hereditarily dominated by classes of complete bipartite graphs, stars, connected bipartite graphs, and complete k-partite graphs.

Infinite versus finite graph domination

Abstract In a graph G with maximum clique size ω, a clique-pair means two cliques of size ω whose intersection is ω − 1. The subject of this paper is the so-called clique-pair conjecture (CPC) which states that if a uniquely colorable perfect graph is not a clique then it contains a clique-pair. We study the structure of the possible counterexamples to this conjecture, and, combining our results with those in Fonlupt and Zemirline (1987), we obtain a new proof of the CPC for 3-chromatic graphs. Theorem 2 states the validity of the CPC for claw-free graphs.

Characterization of graphs dominated by induced paths

Abstract A k -dominating set in a graph G is a set S of vertices such that every vertex of G is at distance at most k from some vertex of S . Given a class D of finite simple graphs closed under connected induced subgraphs, we completely characterize those graphs G in which every connected induced subgraph has a connected k -dominating subgraph isomorphic to some D ∈ D . We apply this result to prove that the class of graphs hereditarily D -dominated within distance k is the same as the one obtained by iteratively taking the class of graphs hereditarily dominated by the previous class in the iteration chain. This strong relation does not remain valid if the initial hereditary restriction on D is dropped.

THE COST CHROMATIC NUMBER AND HYPERGRAPH PARAMETERS

We raise the following general problem: Which structural properties of dominating subgraphs in finite graphs remain valid for infinite graphs? Positive and negative results are presented.

Elementary Bipartite Graphs and Unique Colourability

We give a characterization, in terms of forbidden induced subgraphs, of those graphs in which every connected induced subgraph has a dominating induced path on at most k vertices (k>=3). We show, in particular, that k=4 means precisely the class of domination-reducible graphs, whose original definition applied four types of structural reduction.