Generalized Planar Turán Numbers

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

On weighted efficient total domination

The VERTEX COLOURING problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K3, F)-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices the VERTEX COLOURING problem is polynomial-time solvable in the class of (K3, F)-free graphs. In the present paper, we show that the problem is also polynomial-time solvable in many classes of (K3, F)-free graphs with F being a forest on 6 vertices.

Several problems, including the maximum integral two-flow problem, are known to be NP-complete, but efficiently solvable for planar graphs. In this paper, we extend the algorithm for maximum integral two-flow in planar graphs to certain undirected K 3,3 -free graphs (graphs not containing any subgraph homeomorphic to K 3,3 ). For such graphs, we provide an O(n) algorithm for determining a maximum integral two-flow of a value not less than the value of a maximum two-flow minus two.

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5,Dm)-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.

In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk et al. (55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, 2014) gave a polynomial kernel for Steiner Tree in planar graphs and graphs of bounded genus, when parameterized by $$|T|+k$$ , the total number of vertices in the constructed subgraph. In this paper we present several polynomial time applicable reduction rules for Steiner Tree in graphs of bounded genus. In an instance reduced with respect to the presented reduction rules, the number of terminals |T| is at most cubic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in graphs of bounded genus for the parameterization by the number k of Steiner vertices in the solution. We give better bounds for Steiner Tree in planar graphs.

We show that for an integer k ≥ 2 and an n-vertex graph G without a K 3,3 (resp., K 5) minor, we can compute k induced subgraphs of G with treewidth ≤ 3k−4 (resp., ≤ 6k−7) in O(kn) (resp., O(kn+n 2)) time such that each vertex of G appears in exactly k − 1 of these subgraphs. This leads to practical polynomial-time approximation schemes for various maximum induced-subgraph problems on graphs without a K 3,3 or K 5 minor. The result extends a well-known result of Baker that there are practical polynomial-time approximation schemes for various maximum induced-subgraph problems on planar graphs.

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this paper, we generalize these results to $$K_{3,3}$$ -minor free ( $$K_{3,3}$$ -subdivision free) graphs.

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C4)‐free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw‐free graphs.

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)−1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K 3,3-minor free graphs (these include all planar graphs as well as many others) in O(n 3.19) worst case time. Our algorithm can also count the number of perfect matchings in K 3,3-minor free graphs in O(n 2.19) time.

Generalized Turán problems ask for the maximum number of copies of a graph $H$ in an $n$-vertex, $F$-free graph, denoted by $ex(n,H,F)$. We show how to extend the new, localized approach of Bradač, Malec, and Tompkins to generalized Turán problems. We weight the copies of $H$ (typically taking $H=K_t$), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of $H$, and in each case prove a tight upper bound on the sum of the weights. The generalized edge Turán number $mex(m,H,F)$ is the maximum number of copies of a graph $H$ in an $m$-edge, $F$-free graph. A consequence of our new localized theorems is an asymptotic determination of $ex(n,H,K_{1,r})$ for every $H$ having at least one dominating vertex and $mex(m,H,K_{1,r})$ for every $H$ having at least two dominating vertices.

We obtain sufficient conditions for the emergence of spanning and almost‐spanning bounded‐degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of , using a palette of size , a.a.s. admits a rainbow copy of any given bounded‐degree tree on at most vertices, where is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded‐degree almost‐spanning prescribed trees in , where is independent of . Given an ‐vertex graph with minimum degree at least , where is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph , using colors, where is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded‐degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that , where is independent of , a.a.s. admits a copy of any given bounded‐degree spanning tree. Finally, and with as above, we prove that a uniform coloring of using colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

Finding dominating induced matchings in [formula omitted]-free graphs in polynomial time

We study the dominating set problem in an online setting. An algorithm is required to guarantee competitiveness against an adversary that reveals the input graph one node at a time. When a node is revealed, the algorithm learns about the entire neighborhood of the node (including those nodes that have not yet been revealed). Furthermore, the adversary is required to keep the revealed portion of the graph connected at all times. We present an algorithm that achieves 2-competitiveness on trees. We also present algorithms that achieve 2.5-competitiveness on cactus graphs, (t-1)-competitiveness on K_{1,t}-free graphs, and Θ(√{Δ}) for maximum degree Δ graphs. We show that all of those competitive ratios are tight. Then, we study several more general classes of graphs, such as threshold, bipartite planar, and series-parallel graphs, and show that they do not admit competitive algorithms (i.e., when competitive ratio is independent of the input size). Previously, the dominating set problem was considered in a different input model (often together with the restriction of the input graph being always connected), where a vertex is revealed alongside its restricted neighborhood: those neighbors that are among already revealed vertices. Thus, conceptually, our results quantify the value of knowing the entire neighborhood at the time a vertex is revealed as compared to the restricted neighborhood. For instance, it was known in the restricted neighborhood model that 3-competitiveness is optimal for trees, whereas knowing the neighbors allows us to improve it to 2-competitiveness.

In this paper we disprove a conjecture of Lidický and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most~$2$ in a triangle-free graph.