81 research outputs found
On graphic and 3-hypergraphic sequences
AbstractIn this paper we give a necessary condition for a sequence π of integers to be 3-hypergraphic. This necessary condition is on the lines of Erdős and Gallai conditions for graphic sequences and depends on a function Mr defined on π
Embedding complete ternary tree in hypercubes using AVL trees
A complete ternary tree is a tree in which every non-leaf vertex has exactly three children. We prove that a complete ternary tree of height h, TTh, is embeddable in a hypercube of dimension . This result coincides with the result of [2]. However, in this paper, the embedding utilizes the knowledge of AVL trees. We prove that a subclass of AVL trees is a subgraph of hypercube. The problem of embedding AVL trees in hypercube is an independent emerging problem
Characterizations of 2-variegated graphs and of 3-variegated graphs
AbstractA graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. We prove that a graph G on 2n vertices is 2-variegated if and only if there exists a set S of n independent edges in G such that no cycle in G contains an odd number of edges from S. We also characterize 3-variegated graphs
Parameterized Inapproximability of Independent Set in -Free Graphs
We study the Independent Set (IS) problem in -free graphs, i.e., graphs
excluding some fixed graph as an induced subgraph. We prove several
inapproximability results both for polynomial-time and parameterized
algorithms.
Halld\'orsson [SODA 1995] showed that for every IS has a
polynomial-time -approximation in -free
graphs. We extend this result by showing that -free graphs admit a
polynomial-time -approximation, where is the
size of a maximum independent set in . Furthermore, we complement the result
of Halld\'orsson by showing that for some there is
no polynomial-time -approximation for these graphs, unless NP = ZPP.
Bonnet et al. [IPEC 2018] showed that IS parameterized by the size of the
independent set is W[1]-hard on graphs which do not contain (1) a cycle of
constant length at least , (2) the star , and (3) any tree with two
vertices of degree at least at constant distance.
We strengthen this result by proving three inapproximability results under
different complexity assumptions for almost the same class of graphs (we weaken
condition (2) that does not contain ). First, under the ETH, there
is no algorithm for any computable function .
Then, under the deterministic Gap-ETH, there is a constant such that
no -approximation can be computed in time. Also,
under the stronger randomized Gap-ETH there is no such approximation algorithm
with runtime .
Finally, we consider the parameterization by the excluded graph , and show
that under the ETH, IS has no algorithm in -free graphs
and under Gap-ETH there is no -approximation for -free
graphs with runtime .Comment: Preliminary version of the paper in WG 2020 proceeding
Structure and algorithms for (cap, even hole)-free graphs
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48
On forcibly connected graphic sequences
AbstractA few sufficient conditions for a graphic sequence to be forcibly connected are obtained
Embedding complete ternary trees into hypercubes
We inductively describe an embedding of a complete ternary tree Tₕ of height h into a hypercube Q of dimension at most ⎡(1.6)h⎤+1 with load 1, dilation 2, node congestion 2 and edge congestion 2. This is an improvement over the known embedding of Tₕ into Q. And it is very close to a conjectured embedding of Havel [3] which states that there exists an embedding of Tₕ into its optimal hypercube with load 1 and dilation 2. The optimal hypercube has dimension ⎡(log₂3)h⎤ ( = ⎡(1.585)h⎤) or ⎡(log₂3)h⎤+1
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