122 research outputs found

### Induced Minor Free Graphs: Isomorphism and Clique-width

Given two graphs $G$ and $H$, we say that $G$ contains $H$ as an induced
minor if a graph isomorphic to $H$ can be obtained from $G$ by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph $H$ that Graph Isomorphism is
polynomial-time solvable on $H$-induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs $H$ for which
$H$-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015

### Sublinear-Space Lexicographic Depth-First Search for Bounded Treewidth Graphs and Planar Graphs

The lexicographic depth-first search (Lex-DFS) is one of the first basic graph problems studied in the context of space-efficient algorithms. It is shown independently by Asano et al. [ISAAC 2014] and Elmasry et al. [STACS 2015] that Lex-DFS admits polynomial-time algorithms that run with O(n)-bit working memory, where n is the number of vertices in the graph. Lex-DFS is known to be P-complete under logspace reduction, and giving or ruling out polynomial-time sublinear-space algorithms for Lex-DFS on general graphs is quite challenging. In this paper, we study Lex-DFS on graphs of bounded treewidth. We first show that given a tree decomposition of width O(n^(1-?)) with ? > 0, Lex-DFS can be solved in sublinear space. We then complement this result by presenting a space-efficient algorithm that can compute, for w ? ?n, a tree decomposition of width O(w ?nlog n) or correctly decide that the graph has a treewidth more than w. This algorithm itself would be of independent interest as the first space-efficient algorithm for computing a tree decomposition of moderate (small but non-constant) width. By combining these results, we can show in particular that graphs of treewidth O(n^(1/2 - ?)) for some ? > 0 admits a polynomial-time sublinear-space algorithm for Lex-DFS. We can also show that planar graphs admit a polynomial-time algorithm with O(n^(1/2+?))-bit working memory for Lex-DFS

### Parameterized Complexity of Graph Burning

Graph Burning asks, given a graph $G = (V,E)$ and an integer $k$, whether
there exists $(b_{0},\dots,b_{k-1}) \in V^{k}$ such that every vertex in $G$
has distance at most $i$ from some $b_{i}$. This problem is known to be
NP-complete even on connected caterpillars of maximum degree $3$. We study the
parameterized complexity of this problem and answer all questions arose by Kare
and Reddy [IWOCA 2019] about parameterized complexity of the problem. We show
that the problem is W[2]-complete parameterized by $k$ and that it does no
admit a polynomial kernel parameterized by vertex cover number unless
$\mathrm{NP} \subseteq \mathrm{coNP/poly}$. We also show that the problem is
fixed-parameter tractable parameterized by clique-width plus the maximum
diameter among all connected components. This implies the fixed-parameter
tractability parameterized by modular-width, by treedepth, and by distance to
cographs. Although the parameterization by distance to split graphs cannot be
handled with the clique-width argument, we show that this is also tractable by
a reduction to a generalized problem with a smaller solution size.Comment: 10 pages, 2 figures, IPEC 202

### Reconfiguration of Colorable Sets in Classes of Perfect Graphs

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration

### Low-Congestion Shortcut and Graph Parameters

Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of Omega~(sqrt{n} + D) rounds for many global problems, where n is the number of nodes and D is the diameter of the input graph. Since such a lower bound is derived from special "hard-core" instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts is initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. Specifically, given a specific graph class X, an f-round algorithm of constructing shortcuts of quality q for any instance in X results in O~(q + f)-round algorithms of solving several fundamental graph problems such as minimum spanning tree and minimum cut, for X. The main interest on this line is to identify the graph classes allowing the shortcuts which are efficient in the sense of breaking O~(sqrt{n}+D)-round general lower bounds.
In this paper, we consider the relationship between the quality of low-congestion shortcuts and three major graph parameters, chordality, diameter, and clique-width. The main contribution of the paper is threefold: (1) We show an O(1)-round algorithm which constructs a low-congestion shortcut with quality O(kD) for any k-chordal graph, and prove that the quality and running time of this construction is nearly optimal up to polylogarithmic factors. (2) We present two algorithms, each of which constructs a low-congestion shortcut with quality O~(n^{1/4}) in O~(n^{1/4}) rounds for graphs of D=3, and that with quality O~(n^{1/3}) in O~(n^{1/3}) rounds for graphs of D=4 respectively. These results obviously deduce two MST algorithms running in O~(n^{1/4}) and O~(n^{1/3}) rounds for D=3 and 4 respectively, which almost close the long-standing complexity gap of the MST construction in small-diameter graphs originally posed by Lotker et al. [Distributed Computing 2006]. (3) We show that bounding clique-width does not help the construction of good shortcuts by presenting a network topology of clique-width six where the construction of MST is as expensive as the general case

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