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Let $P_k$ be a path, $C_k$ a cycle on $k$ vertices, and $K_{k,k}$ a complete bipartite graph with $k$ vertices on each side of the bipartition. We prove that (1) for any integers $k, t>0$ and a graph $H$ there are finitely many subgraph minimal graphs with no induced $P_k$ and $K_{t,t}$ that are not $H$-colorable and (2) for any integer $k>4$ there are finitely many subgraph minimal graphs with no induced $P_k$ that are not $C_{k-2}$-colorable. The former generalizes the result of Hell and Huang [Complexity of coloring graphs without paths and cycles, Discrete Appl. Math. 216: 211--232 (2017)] and the latter extends a result of Bruce, Hoang, and Sawada [A certifying algorithm for 3-colorability of $P_5$-Free Graphs, ISAAC 2009: 594--604]. Both our results lead to polynomial-time certifying algorithms for the corresponding coloring problems.

A well-quasi-order is an order which contains no infinite decreasing sequence and no infinite collection of incomparable elements. In this paper, we consider graph classes defined by excluding one graph as contraction. More precisely, we give a complete characterization of graphs H such that the class of H-contraction-free graphs is well-quasi-ordered by the contraction relation. This result is the contraction analogue on the previous dichotomy theorems of Damsaschke [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427-435, 1990] on the induced subgraph relation, Ding [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489-502, 1992] on the subgraph relation, and B{\l}asiok et al. [Induced minors and well-quasi-ordering, ArXiv e-prints, 1510.07135, 2015] on the induced minor relation., Comment: 18 pages

A graph $H$ is an induced minor of a graph $G$ if it can be obtained from an induced subgraph of $G$ by contracting edges. Otherwise, $G$ is said to be $H$-induced minor-free. Robin Thomas showed that $K_4$-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without $K_4$ and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 -- 247, 1985]. We provide a dichotomy theorem for $H$-induced minor-free graphs and show that the class of $H$-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if $H$ is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990].

Let fvs$(G)$ and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph $G$, respectively. The price of connectivity for feedback vertex set (poc-fvs) for a class of graphs ${\cal G}$ is defined as the maximum ratio $\mbox{cfvs}(G)/\mbox{fvs}(G)$ over all connected graphs $G\in {\cal G}$. We study the poc-fvs for graph classes defined by a finite family ${\cal H}$ of forbidden induced subgraphs. We characterize exactly those finite families ${\cal H}$ for which the poc-fvs for ${\cal H}$-free graphs is upper bounded by a constant. Additionally, for the case where $|{\cal H}|=1$, we determine exactly those graphs $H$ for which there exists a constant $c_H$ such that $\mbox{cfvs}(G)\leq \mbox{fvs}(G) + c_H$ for every connected $H$-free graph $G$, as well as exactly those graphs $H$ for which we can take $c_H=0$.

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a non-negative integer $k,$ decide whether the cyclability of $G$ is at least $k,$ is {\sf NP}-hard. We study the parametrized complexity of this problem. We prove that this problem, parameterized by $k,$ is ${\sf co\mbox{-}W[1]}$-hard and that its does not admit a polynomial kernel on planar graphs, unless ${\sf NP}\subseteq{\sf co}\mbox{-}{\sf NP}/{\sf poly}$. On the positive side, we give an {\sf FPT} algorithm for planar graphs that runs in time $2^{2^{O(k^2\log k)}}\cdot n^2$. Our algorithm is based on a series of graph-theoretical results on cyclic linkages in planar graphs.

We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental.

We consider the problem Scattered Cycles which, given a graph $G$ and two positive integers $r$ and $\ell$, asks whether $G$ contains a collection of $r$ cycles that are pairwise at distance at least $\ell$. This problem generalizes the problem Disjoint Cycles which corresponds to the case $\ell = 1$. We prove that when parameterized by $r$, $\ell$, and the maximum degree $\Delta$, the problem Scattered Cycles admits a kernel on $24 \ell^2 \Delta^\ell r \log(8 \ell^2 \Delta^\ell r)$ vertices. We also provide a $(16 \ell^2 \Delta^\ell)$-kernel for the case $r=2$ and a $(148 \Delta r \log r)$-kernel for the case $\ell = 1$. Our proofs rely on two simple reduction rules and a careful analysis., Comment: 17 pages

Suppose that we are given two independent sets $I_0$ and $I_r$ of a graph such that $|I_0| = |I_r|$, and imagine that a token is placed on each vertex in $I_0$. The token jumping problem is to determine whether there exists a sequence of independent sets which transforms $I_0$ into $I_r$ so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. This problem is known to be PSPACE-complete even for planar graphs of maximum degree three, and W[1]-hard for general graphs when parameterized by the number of tokens. In this paper, we present a fixed-parameter algorithm for the token jumping problem on planar graphs, where the parameter is only the number of tokens. Furthermore, the algorithm can be modified so that it finds a shortest sequence for a yes-instance. The same scheme of the algorithms can be applied to a wider class of graphs, $K_{3,t}$-free graphs for any fixed integer $t \ge 3$, and it yields fixed-parameter algorithms.

We present a polynomial-time algorithm that, given two independent sets in a claw-free graph $G$, decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex $v$ from the current independent set $S$ and to add a new vertex $w$ (not in $S$) such that the result is again an independent set. We also consider the more restricted model where $v$ and $w$ have to be adjacent.

Given two finite posets P and Q, P is a chain minor of Q if there exists a partial function f from the elements of Q to the elements of P such that for every chain in P there is a chain C_Q in Q with the property that f restricted to C_Q is an isomorphism of chains. We give an algorithm to decide whether a poset P is a chain minor of o poset Q that runs in time O(|Q| log |Q|) for every fixed poset P. This solves an open problem from the monograph by Downey and Fellows [Parameterized Complexity, 1999] who asked whether the problem was fixed parameter tractable.

We prove a structural characterization of graphs that forbid a fixed graph $H$ as an immersion and can be embedded in a surface of Euler genus $\gamma$. In particular, we prove that a graph $G$ that excludes some connected graph $H$ as an immersion and is embedded in a surface of Euler genus $\gamma$ has either "small" treewidth (bounded by a function of $H$ and $\gamma$) or "small" edge connectivity (bounded by the maximum degree of $H$). Using the same techniques we also prove an excluded grid theorem on bounded genus graphs for the immersion relation.

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors. Vizing's Theorem gives a bound of $\frac{\Delta}{\Delta+1}|E|$. This is known to be tight for cliques $K_{\Delta+1}$ when $\Delta$ is even. However, for $\Delta=3$ it was improved to $26/31|E|$ by Albertson and Haas [Parsimonious edge colorings, Disc. Math. 148, 1996] and later to $6/7|E|$ by Rizzi [Approximating the maximum 3-edge-colorable subgraph problem, Disc. Math. 309, 2009]. It is tight for $B_3$, the graph isomorphic to a $K_4$ with one edge subdivided. We improve previously known bounds for $\Delta\in{3,...,7}$, under the assumption that for $\Delta=3,4,6$ graph $G$ is not isomorphic to $B_3$, $K_5$ and $K_7$, respectively. For $\Delta \geq 4$ these are the first results which improve over the Vizing's bound. We also show a new bound for subcubic multigraphs not isomorphic to $K_3$ with one edge doubled. In the second part, we give approximation algorithms for the Maximum k-Edge-Colorable Subgraph problem, where given a graph G (without any bound on its maximum degree or other restrictions) one has to find a k-edge-colorable subgraph with maximum number of edges. In particular, when G is simple for k=3,4,5,6,7 we obtain approximation ratios of 13/15, 9/11, 19/22, 23/27 and 22/25, respectively. We also present a 7/9-approximation for k=3 when G is a multigraph. The approximation algorithms follow from a new general framework that can be used for any value of k., Comment: 34 pages

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting from graphs that are planar sub-cubic or of branch-width at most 10.

For every graph $H$, there exists a polynomial-time algorithm deciding if a planar input graph $G$ can be contracted to~$H$. However, the degree of the polynomial depends on the size of $H$. In this paper, we identify a class of graphs $\cal C$ such that for every $H \in \cal C$, there exists an algorithm deciding in time $f(|V(H)|) \cdot |V(G)|^{\bigO{1}}$ whether a planar graph $G$ can be contracted to~$H$. (The function $f(\cdot)$ does not depend on $G$.) The class $\cal C$ is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph $H \in \cal C$ if and only if there exists a constant $c_H$ such that if the tree-width of a graph is at least $c_H$, it contains $H$ as a contraction. We also provide a characterization of $\cal C$ in terms of minimal forbidden contractions., Comment: 11 pages, 3 figues

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships between these models, observing that shortest path reconfiguration is a special case of independent set reconfiguration in perfect graphs, under any of the three models. Finally, we give polynomial results for restricted classes of graphs (even-hole-free and $P_4$-free graphs).

Testing whether there is an induced path in a graph spanning k given vertices is already NP-complete in general graphs when k=3. We show how to solve this problem in polynomial time on claw-free graphs, when k is not part of the input but an arbitrarily fixed integer., Comment: 12 pages, 1 figure, STACS 2010