Unique factorisation of additive induced-hereditary properties

Minimal reducible bounds for induced-hereditary properties

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property P is of finite character if a graph G has a property P if and only if every finite induced subgraph of G has a property P . Let P1,P2, . . . ,Pn be graph properties of finite character, a graph G is said to be (uniquely) (P1,P2, . . . ,Pn)partitionable if there is an (exactly one) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. Let us denote by R = P1◦P2◦ · · · ◦Pn the class of all (P1,P2, . . . ,Pn)-partitionable graphs. A property R = P1◦P2◦ · · · ◦Pn, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P1,P2, . . . ,Pn)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property R of finite character there exists a weakly universal countable graph if and only if each property Pi has a weakly universal graph. 242 J. Bucko and P. Mihok

The maximum number of complete subgraphs in a graph with given maximum degree

In graph theory, a “dominating vertex set” is a subset of vertices in a graph such that every vertex in the graph is either a member of the subset or adjacent to a member of the subset. In other words, the vertices in the dominating set “dominate” the remaining vertices in the graph. Dominating vertex sets are important in graph theory because they can help us understand and analyze the behavior of a graph. For example, in network analysis, a set of dominant vertices may represent key nodes in a network that can influence the behavior of other nodes. Identifying dominant sets in a graph can also help in optimization problems, as it can help us find the minimum set of vertices that can control the entire graph. Now that there are theories about vagueness, it is important to define parallel ideas in vague structures, such as intuitionistic fuzzy graphs. This paper describes a better way to find dominating vertex sets (DVSs) in intuitive fuzzy graphs (IFGs). Even though there is already an algorithm for finding DVSs in IFGs, it has some problems. For example, it does not take into account the vertex volume, which has a direct effect on how DVSs are calculated. To address these limitations, we propose a new algorithm that can handle large-scale IFGs more efficiently. We show how effective and scalable the method is by comparing it to other methods and applying it to water flow. This work’s contributions can be used in many areas, such as social network analysis, transportation planning, and telecommunications.

In the paper we introduce the new game--the unilateral $${\mathcal{P}}$$ -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property $${\mathcal {P}}$$ and a set C of r colours. In the unilateral $${\mathcal {P}}$$ -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral $${\mathcal {P}}$$ -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property $${\mathcal {P}}$$ . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The $${\mathcal {P}}$$ -unilateral game chromatic number, denoted by $${\chi_{ug}^\mathcal {P}(G)}$$ , is the least number r for which Alice has a winning strategy for the unilateral $${\mathcal {P}}$$ -colouring game with r colours on G. We prove that the $${\mathcal {P}}$$ -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral $${\mathcal {P}}$$ -colouring game. Moreover, for k ≥ 2 we define a class of graphs $${\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}$$ . The class $${\mathcal {H}_k }$$ contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let $${\mathcal {S}_d}$$ be the class of graphs with maximum degree at most d. We find the upper bound for the $${\mathcal {S}_2}$$ -unilateral game chromatic number for graphs from $${\mathcal {H}_3}$$ and we study the $${\mathcal {S}_d}$$ -unilateral game chromatic number for graphs from $${\mathcal {H}_4}$$ for $${d \in \{2,3\}}$$ . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if $${G \in \mathcal {H}_k}$$ , then $${\chi_g^{(d)}(G) \leq k + 2-d}$$ , for $${k \in \{3, 4\}}$$ and $${d \in \{0, \ldots, k-1\}}$$ . This generalizes a known result for trees.

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

A jump to the bell number for hereditary graph properties

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it is shown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

: There are four prominent product graphs in graph theory: Cartesian, strong, direct and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted G o H, for some digraphs G and H. The paper concludes by proving a cancellation property for the lexicographic product of digraphs G, H, A, and B: if G o H = A o B and /V(G)/ = /V(A)/, then G = A. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.

The workshop provided a venue for discussing several recent major developments in graph theory, with a primary focus on results concerning the notion of twin-width and hereditary properties of graphs. Both areas have seen a very rapid development recently, as reflected in the many results presented during the workshop. In addition to many interesting talks spanning the whole breadth of graph theory, the workshop also offered valuable collaboration opportunities, which also engaged early career researchers.

In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemeredi regularity, which is a strong, approximate version of this property that relates to pseudorandomness. We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory. We then turn to a form of Szemeredi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs. Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles.

The problem of finding a simple regular graph with the specified order and degree that minimizes the average shortest path length has a long history in graph theory. Recently this problem has attracted a great deal of attention in relation to the design of computer networks in data centers. In this paper, we propose a genetic algorithm for finding an approximate solution to this problem. Because the search space is the set of all simple regular graphs with the specified order and degree, conventional genetic algorithms cannot be directly applied. We propose in this paper new crossover and mutation operators that guarantee the simplicity and regularity of graphs. We also evaluate the effectiveness of the proposed method experimentally.