Computer Science

In this paper, we prove that there is a strongly universal cellular automaton on the heptagrid with seven states which is rotation invariant. This improves a previous paper of the author where the automaton required ten states.

A classical theorem of Nisan and Szegedy says that a boolean function with degree d as a real polynomial depends on at most d 2 d−1 of its variables. In recent work by Chiarelli, Hatami and Saks, this upper bound was improved to C⋅ 2 d , where C=6.614 . Here we refine their argument to show that one may take C=4.416 .

In this third paper, we revisit the question to which extent the properties of the trees associated to the tilings {p,4} of the hyperbolic plane are still true if we consider a finitely generated tree by the same rules but rooted at a black node? What happens if, considering the same distinction between black and white nodes but changing the place of the black son in the rules. What happens if we change the representation of the numbers by another set of digits? We tackle all of these questions in the paper. The present paper is an extension of the previous papers arXiv:1904.12135 and arXiv:1907.04677.

In this second paper, we look at the following question: are the properties of the trees associated to the tilings {p,4} and {p + 2,3} of the hyperbolic plane still true if we consider a finitely generated tree by the same rules but rooted at a black node? The direct answer is no, but new properties arise, no more complex than in the case of a tree rooted at a white node, and worth of interest. The present paper is an extension of the previous paper: arXiv:1904.12135.

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{č}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree Δ is acyclically edge (Δ+2) -colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1) -Labelling). A classical complexity result on Colouring is a well-known dichotomy for H -free graphs (a graph is H -free if it does not contain H as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on H -free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours k is fixed, that is, not part of the input. From our study it follows that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

Let G(V,E) be a simple, undirected and connected graph. A dominating set S⊆V(G) is called a 2 -\textit{secure dominating set} ( 2 -SDS) in G , if for every pair of distinct vertices u 1 , u 2 ∈V(G) there exists a pair of distinct vertices v 1 , v 2 ∈S such that v 1 ∈N[ u 1 ] , v 2 ∈N[ u 2 ] and (S∖{ v 1 , v 2 })∪{ u 1 , u 2 } is a dominating set in G . The 2 \textit{-secure domination number} denoted by γ 2s (G) , equals the minimum cardinality of a 2 -SDS in G . Given a graph G and a positive integer k, the 2 -Secure Domination ( 2 -SDM) problem is to check whether G has a 2 -secure dominating set of size at most k. It is known that 2 -SDM is NP-complete for bipartite graphs. In this paper, we prove that the 2 -SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs. We strengthen the NP-complete result for bipartite graphs, by proving this problem is NP-complete for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite graphs. We prove that 2 -SDM is linear time solvable for bounded tree-width graphs. We also show that the 2 -SDM is W[2]-hard even for split graphs. The Minimum 2 -Secure Dominating Set (M2SDS) problem is to find a 2 -secure dominating set of minimum size in the input graph. We propose a Δ(G)+1 − approximation algorithm for M2SDS, where Δ(G) is the maximum degree of the input graph G and prove that M2SDS cannot be approximated within (1−ϵ)ln(|V|) for any ϵ>0 unless NP⊆DTIME(|V | O(loglog|V|) ) . % even for bipartite graphs. A secure dominating set of a graph \textit{defends} one attack at any vertex of the graph. Finally, we show that the M2SDS is APX-complete for graphs with Δ(G)=4.

Let G=(V,E) be a simple, undirected and connected graph. A connected dominating set S⊆V is a secure connected dominating set of G , if for each u∈V∖S , there exists v∈S such that (u,v)∈E and the set (S∖{v})∪{u} is a connected dominating set of G . The minimum size of a secure connected dominating set of G denoted by γ sc (G) , is called the secure connected domination number of G . Given a graph G and a positive integer k, the Secure Connected Domination (SCDM) problem is to check whether G has a secure connected dominating set of size at most k. In this paper, we prove that the SCDM problem is NP-complete for doubly chordal graphs, a subclass of chordal graphs. We investigate the complexity of this problem for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a secure connected dominating set of minimum size in the input graph. We propose a (Δ(G)+1) - approximation algorithm for MSCDS, where Δ(G) is the maximum degree of the input graph G and prove that MSCDS cannot be approximated within (1−ϵ)ln(|V|) for any ϵ>0 unless NP⊆DTIME(|V | O(loglog|V|) ) even for bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs with Δ(G)=4 .

A set S⊆V is a dominating set in G if for every u \in V \ S, there exists v∈S such that (u,v)∈E , i.e., N[S]=V . A dominating set S is an Isolate Dominating Set} (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S⊆V is an independent set if G[S] has no edge. A set S \subseteq V is a secure dominating set of G if, for each vertex u∈V∖S , there exists a vertex v∈S such that (u,v)∈E and (S {v})∪{u} is a dominating set of G . In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S⊆V is an Independent Secure Dominating Set (InSDS) if S is an independent set and a secure dominating set of G . The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γ is (G) . Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we prove that the MInSDS problem is APX-hard for graphs with maximum degree 5.

A dominating set S is an Isolate Dominating Set (IDS) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S⊆V is an isolate secure dominating set (ISDS), if for each vertex u∈V∖S , there exists a neighboring vertex v of u in S such that (S∖{v})∪{u} is an IDS of G . The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by γ 0s (G) . Given a graph G=(V,E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.