Richard J. Nowakowski

Vertex-to-vertex pursuit in a graph

A graph G is given and two players, a cop and a robber, play the following game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop. A move consists of staying at ones present vertex or moving to an adjacent vertex; each move is seen by both players. The cop wins if he manages to occupy the same vertex as the robber, and the robber wins if he avoids this forever. We characterize the graphs on which the cop has a winning strategy, and connect the problem with the structure theory of graphs based on products and retracts.

The game of cops and robbers on graphs

This book is the first and only one of its kind on the topic of Cops and Robbers games, and more generally, on the field of vertex pursuit games on graphs. The book is written in a lively and highly readable fashion, which should appeal to both senior undergraduates and experts in the field (and everyone in between). One of the main goals of the book is to bring together the key results in the field; as such, it presents structural, probabilistic, and algorithmic results on Cops and Robbers games. Several recent and new results are discussed, along with a comprehensive set of references. The book is suitable for self-study or as a textbook, owing in part to the over 200 exercises. The reader will gain insight into all the main directions of research in the field and will be exposed to a number of open problems.

A characterization of well covered graphs of girth 5 or greater

Abstract A graph is well covered if every maximal independent set has the same cardinality. A vertex x , in a well covered graph G , is called extendable if G − { x } is well covered and β( G ) = β( G − { x }). If G is a connected, well covered graph of girth ≥ 5 and G contains an extendable vertex then G is the disjoint union of edges and 5-cycles together with a restricted set of edges joining these subgraphs. There are only six connected, well covered graphs of girth ≥ 5 which do not contain an extendable vertex.

Associative graph products and their independence, domination and coloring numbers

Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗ and parameter p pairs are multiplicative, that is, p(G ⊗H) ≥ p(G)p(H) for all graphs G and H or p(G⊗H) ≤ p(G)p(H) for all graphs G and H. The parameters are related to independence, domination and irredundance. This includes Vizing’s conjecture directly, and indirectly the Shannon capacity of a graph and Hedetniemi’s coloring conjecture.

Roots of Independence Polynomials of Well Covered Graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ∑ ikxk. In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤ β (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each β) for which the independence polynomials have a root arbitrarily close to −β.

On the Location of Roots of Independence Polynomials

The independence polynomial of a graph G is the function i(G, x) = ∑k≥0ikxk, where ik is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (−∞, 0], while complex roots are dense in ℂ, even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.

The smallest graph variety containing all paths

This paper is inspired by two problems. Chatuctetize the retpacts of a graph. In this respect this paper continues the investigations of [2-6]. Ck&fi gruphs u.ccor&ng to their retracts. In this respect this paper begins the classification theory for graphs, initiated by D. Duffus and I. Eva1 in [l] for ordered sets. For a graph G let V(G) denote its vertex set and E(G) E V(G) x V(G) its edge set. A graph H is a retract of the graph G if there are edge-preserving maps, f of V(H) to V(G).. and g of V(G) to V(H), satisfying g 0 f(u) = 27 for each v E V(H). Our interest in this paper is with the retracts of a reflexive graph-an undirected graph without any multiple edges but with a loop at every vertex (cf. Fig. l(a)). In [2] we have shown, for instance, that in a reflexive graph every cycle of minimum order and e:uery isometric tree is a retract. For graphs G and H the direct product G x H is the graph with vertex set V(G) x V(H) and edge set consisting of all pairs ((a, x), (6, y)) where (a, 6) E E(G) and (x, y)~ .E(H) (cf. Fig. l(b)). A representation of a reflexive graph G is a family (Gi 1 i E I) of reflexive graphs such that each Gi is a retract of G and G is itself a retract of the direct product ni,, Gi. (See Fig. 2.) G is imeducible if, for every representation (Gi 1 i E I) of G, G is a retract of Gi for some i E I; otherwise, G is reducible. A path P is a graph whose vertex set consists of a sequence uo, u1, u2, ’ , %, l ’ of distinct vertices and (a,, &) E E(P), for each i = 1,2,... ; the length I(P) of a finite path P = (uO, al, u2, . . . , a,,:) is a, where n 3 1.

A characterization of well-covered graphs that contain neither 4- nor 5-cycles

A graph is well covered if every maximal independent set has the same cardinality. A vertex x, in a well-covered graph G, is called extendable if G – {x} is well covered and β(G) = β(G – {x}). If G is a connected, well-covered graph containing no 4- nor 5-cycles as subgraphs and G contains an extendable vertex, then G is the disjoint union of edges and triangles together with a restricted set of edges joining extendable vertices. There are only 3 other connected, well-covered graphs of this type that do not contain an extendable vertex. Moreover, all these graphs can be recognized in polynomial time.

A game of cops and robbers played on products of graphs

Abstract The game of cops and robbers is played with a set of ‘cops’ and a ‘robber’ who occupy some vertices of a graph. Both sides have perfect information and they move alternately to adjacent vertices. The robber is captured if at least one of the cops occupies the same vertex as the robber. The problem is to determine on a given graph, G , the least number of cops sufficient to capture the robber, called the cop-number, c ( G ). We investigate this game on three products of graphs: the Cartesian, categorical, and strong products.

Extensions and restrictions of Wythoff's game preserving its P positions

We consider extensions and restrictions of Wythoffs game having exactly the same set of P positions as the original game. No strict subset of rules gives the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding numeration system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively P positions of Wythoffs game and moves that can be adjoined.